Unified Framework for Generalized Statistics: Canonical Partition Function, Maximum Occupation Number, and Permutation Phase of Wave Function

Abstract

Over the past decades, many kinds of generalized statistics are proposed through two approaches: (1) generalizing the permutation symmetry of the wave function and (2) generalizing the maximum occupation of the quantum state. Nevertheless, the connection between these two approaches is obscure. In this paper, we suggest a unified framework to describe various kinds of generalized statistics by using the representation theory of the permutation group and the unitary group. With this approach, we reveal the connection between the permutation phase and the maximum occupation number, through constructing a method to obtain the permutation phase and the maximum occupation number from the canonical partition function. We show that only bosonic and fermionic particles are completely indistinguishable under permutations. Particles obeying generalized statistics are not completely indistinguishable and thus are not quantum particles. Besides, we also give the following results: (1) providing a general formula of canonical partition functions of ideal N-particle gases who obey various kinds of generalized statistics, (2) revealing that the maximum occupation number is not sufficient to distinguish different kinds of generalized statistics, (3) specifying the permutation phases of wave functions for generalized statistics, and (4) proposing three new kinds of generalized statistics which seem to be the missing pieces in the puzzle.

1 Introduction

The principle of indistinguishability requires that exchanging two identical particles does not lead to any observable effect [1,2,3].

In quantum mechanics, a physical system is described by a complex wave function, but the observable is a real number. The principle of indistinguishability allows a change on wave functions after exchanging two identical particles so long as the observable does not change. Consequently, the wave function may change a phase factor after exchanging identical particles. It comes naturally Bose–Einstein statistics and Fermi–Dirac statistics whose phase factors change $e^{i0}=1$ and $e^{i\pi }=-1$, respectively. Besides Bose–Einstein statistics and Fermi–Dirac statistics, however, one can still consider other kinds of intermediate statistics so long as it does not violate the principle of indistinguishability, i.e., there are no changes on observables after exchanging identical particles. Generalizing intermediate statistics along this line is to consider phase factors between 0 and $\pi$, e.g., anyons are successful in explaining the fractional quantum Hall effect [4, 5].

In statistical mechanics, macroscopic systems are treated averagely. The number of microstates is the key in the calculation of the average value. Particles are indistinguishable, so exchanging particles occupying different states does not lead to new microstates. In intermediate statistics allowed by the principle of indistinguishability, there is no new microstate after exchanging identical particles. From the view of statistical mechanics, the difference between various intermediate statistics is reflected in the maximum occupation numbers, e.g., for Fermi–Dirac statistics the maximum occupation number is 1 and for Bose–Einstein statistics there is no limitation on the maximum occupation number. Generalizing intermediate statistics along this line is to generalize the maximum occupation number to an arbitrary number, e.g., the spin wave satisfies Gentile statistics [6].

The above analyses shows two approaches of constructing generalized statistics: (1) in quantum mechanics, generalize the permutation symmetry of the wave function and (2) in statistical mechanics, generalize the maximum occupation number.

Over the past decades, many kinds of generalized statistics are proposed along these two approaches, such as parastatistics proposed in 1952 by Green [7, 8], immannons proposed by Tichy in 2017 [9], and Gentileonic statistics proposed by Cattani and Fernandes in 1984 [10] are generalized from permutation symmetry of the wave function. The intermediate statistics or Gentile statistics proposed in 1940 by Gentile Jr [11, 12] and a series of intermediate statistics discussed in [13] are generalized from the maximum occupation number.

It shows that elementary excitations may obey generalized statistics. For example, Gentile statistics can describe the spin wave better [6] and Para-statistics are candidates to be associated with dark matter and/or dark energy [14]. There are discussions on various kinds of generalized statistics. For example, the operator realization of Gentile statistics is given in Refs. [15, 16] and the statistical distribution of various intermediate statistics is calculated from operator relations in Ref. [13]. The generalized statistics such as parastatistics and Gentile statistics are discussed in Ref. [17]. The distinctions between the intermediate statistics, parastatistics, and Okayama statistics are discussed in Ref. [18]. The connection between the irreducible representation of $S_{N}$ and the parastatistics is given by Okayama [19]. The relation between properties of Gentile statistics and the fractional statistics of anyons is discussed in Ref. [20]. Statistical distributions for generalized ideal gases and fractional-statistics gases are given in Ref. [21].

Nevertheless, the connection between various kinds of statistics generalized from those two approaches, the permutation symmetry of the wave function and the maximum occupation number, is obscure.

In this paper, based on the mathematical theory of the invariant matrix [22], the Schur-Weyl duality [23], and the symmetric function [22, 24], we suggest a unified framework to describe various kinds of generalized statistics, including parastatistics [7, 8], Gentile statistics [11, 12], Gentileonic statistics [10], and immannons [9]. With this approach, we reveal the connection between the permutation phase of the wave function and the maximum occupation number, through constructing a method to obtain the permutation phase and the maximum occupation number from the canonical partition function. We show that under the action of permutations, only Bose and Fermi statistics are quantum statistics. Particles obeying generalized statistics are not completely indistinguishable. Concretely, for particles obeying generalized statistics, exchanging two particles occupying different quantum states will lead to either new microstates or the result that Hamiltonian is noninvariant under permutations. Thus, particles obeying generalized statistics are not quantum particles. Therefore, generalized statistics are not quantum statistics.

Besides, we also give the following results: (1) providing a general formula of canonical partition functions of ideal N-particle gases who obey various kinds of generalized statistics, (2) revealing that the maximum occupation number is not sufficient to distinguish different kinds of generalized statistics, (3) specifying the permutation phases of wave functions for generalized statistics, and (4) proposing three new kinds of generalized statistics which seem to be the missing pieces in the puzzle.

This paper is organized as follows. In Sect. 2, we give general formulae of the canonical partition function of an ideal N-particle gas. In Sect. 3, we give a method to obtain the maximum occupation number, the indistinguishability of particles, the permutation phase of the wave function, and the commutation relation between the Hamiltonian and the permutation group directly from the canonical partition. In Sect. 4, we reveal the connection between the permutation phase of the wave function and the maximum occupation number. In Sect. 5, we give a unified framework to describe a series of generalized statistics. The canonical partition function, the maximum occupation number, and the permutation phase of the wave function are given. The intermediate statistics such as parastatistics, Gentile statistics, the immanonns, Gentileonic statistics, and the new proposed generalized statistics are discussed as examples. In Sect. 6, the conclusion and the outlook are given. A brief review of the mathematical theory involved in the present paper and some details of the calculation are given in appendixes.

2 General Formulae of the Canonical Partition Function

In statistical mechanics, a system with the fixed particle number should be considered in the canonical ensemble and the canonical partition function is the key because the thermodynamic information is embedded in the canonical partition function. For example, the eigenvalue spectrum can be obtained from the canonical partition function [25]. However, to calculate the canonical partition function is difficult because one has to deal with the inter-particle interactions and at the same time take the constraint of fixed particle number into consideration. For example, the previous work [26] gives the canonical partition function for ideal Bose, Fermi, and Gentile statistics.

In this section, by using the mathematical theory such as the symmetric function and the invariant matrix, we find that the canonical partition function of an ideal gas can be written as linear combinations of the M-function (the monomial symmetric polynomial, here, for the sake of convenience, we denote it by the M-function) or the S-function (the Schur-function). The M-function and the S-function are two important kinds of symmetric functions [22, 24]. In the present paper, one can treat them as special functions. A brief review of the symmetric function is given in Sect. 7.1.2.

2.1 Expressing the Canonical Partition Function by the M-Function

Usually, to calculate the canonical partition function of an ideal gas is difficult. The difficulty mainly comes from the constraint that the number of particles in the system is fixed. For example, for a given set of the occupation number $\left( a\right) =\left( a_{1},a_{2},\ldots ,a_{i},\ldots \right)$ with $a_{i}$ the number of particles occupying the quantum state i, the canonical partition function, by definition [1, 2], can be written as

$$\begin{aligned} Z\left( \beta ,N\right) =\sum _{\left( a\right) }\Omega ^{\left( a\right) } e^{-\beta \sum _{i}a_{i}\varepsilon _{i}}, \end{aligned}$$

(2.1)

where $\varepsilon _{i}$ is the energy of the quantum state i, $\sum _{i}a_{i}\varepsilon _{i}$ gives the energy of the system, $\Omega ^{\left( a\right) }$ is the number of microstates for a given set of the occupation number $\left( a\right)$. The number of particles in the system is fixed, thus $\sum _{\left( a\right) }$ in Eq. (2.1) is the summation over all possible sets of occupation numbers $\left( a\right)$ subjected to the constraint

$$\begin{aligned} \sum _{i}a_{i}=N. \end{aligned}$$

(2.2)

The constraint about the particle number, Eq. (2.2), makes the calculation of the partition function, Eq. (2.1) difficult.

In this section, by using the mathematical theory of the symmetric function, we calculate the canonical partition function, Eq. (2.1), directly. We show that the canonical partition function of an ideal gas can be written as linear combinations of the M-function.

To calculate Eq. (2.1), firstly, we transform a set of the occupation number $\left( a\right) =\left( a_{1},a_{2},\ldots ,a_{i},\ldots \right)$ into an integer partition $\left( \uplambda \right) =\left( \uplambda _{1},\uplambda _{2},\ldots ,\uplambda _{l} \right)$ of N by equaling $\uplambda _{1}$ with the largest elements in $\left( a\right)$, $\uplambda _{2}$ with the second largest elements in $\left( a\right)$, and so on. An integer partition of N is a way of writing N as sums of other integers. For example, $\left( \uplambda \right) = (2,1)$ (in the integer partition, the element is arranged in descending order) is an integer partition of 3. Then the canonical partition function can be written as

$$\begin{aligned} Z\left( \beta ,N\right) =\sum _{\left( \uplambda \right) }\Omega ^{ \left( \uplambda \right) } \sum _{perm}e^{-\beta \varepsilon _{1}\uplambda _{1} }e^{-\beta \varepsilon _{2}\uplambda _{2}}\ldots . \end{aligned}$$

(2.3)

The trick lies in $\sum _{perm}$ of Eq. (2.3) which indicates that the summation runs over all possible monotonically increasing permutations of $e^{-\beta \varepsilon _{i}}$. $\Omega ^{ \left( \uplambda \right) }$ given by $\Omega ^{ \left( a\right) }$ is the number of microstates corresponding to the integer partition $\left( \uplambda \right)$. By the definition of the M-function [22, 24], we find that $\sum _{perm}e^{-\beta \varepsilon _{1}\uplambda _{1}}e^{-\beta \varepsilon _{2}\uplambda _{2}}\ldots$ is a M-function with variables $x_{i}=e^{-\beta \varepsilon _{i}}$, i.e., $m_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) = \sum _{perm}e^{-\beta \varepsilon _{1}\uplambda _{1}}e^{-\beta \varepsilon _{2}\uplambda _{2}}\ldots$.

Therefore, the canonical partition function of an ideal gas, Eq. (2.3), can be written as

$$\begin{aligned} Z\left( \beta ,N\right) =\sum _{I=1}^{P\left( N\right) }\Omega ^{\left( \uplambda \right) _I}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(2.4)

where we denote $\left( \uplambda \right) _{I}$ the Ith integer partition of N (the order of integer partitions of N is given in Sect. 7.1.1). The summation $\sum _{\left( \uplambda \right) }$ is replaced by $\sum _{I=1}^{P\left( N\right) }$, because to sum over all integer partition of N is equavilent to sum from the first to the last integer partition of N and the last is the $P\left( N\right) th$ integer partition with $P\left( N\right)$ the number of integer partition function of N. For example, $P\left( 3\right) =3$ and to sum over all integer partition of 3 is equavilent to sum from $\left( \uplambda \right) _{1}=\left( 3\right)$, $\left( \uplambda \right) _{2}=\left( 2,1\right)$, and $\left( \uplambda \right) _{3}=\left( 1^{3}\right)$ with $1^{3}$ representing that 1 appears 3 times. $m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$ is the M-function labeled by the Ith integer partition of N . The coefficient $\Omega ^{\left( \uplambda \right) _I}$ is the number of microstates corresponding to the Ith integer partition.

The canonical partition of ideal gases, Eq. (2.4), is one of the first main result in this work and we will show that the coefficient of the M-function $\Omega ^{\left( \uplambda \right) _I}$ in Eq. (2.4) carries important information such as the maximum occupation number and the indistinguishability of the particle in Sect. 3.

Examples: Bose, Fermi, and Gentile statistics. For an ideal Bose gas, when giving a specified set of the occupation number $\left( a\right)$, the number of microstates $\Omega ^{ \left( a\right) }$ is 1 because bosons are indistinguishable and exchanging two bosons that occupy different quantum states will not lead to a new microstate. Therefore, coefficients $\Omega ^{\left( \uplambda \right) _I}$ in Eq. (2.4) are all 1 and the canonical partition function for an ideal Bose gas is

$$\begin{aligned} Z^{B}\left( \beta ,N\right) =\sum _{I=1}^{P\left( N\right) }m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(2.5)

For the sake of clarity, we give the explicit form of Eq. (2.5) for $N=3$:

$$\begin{aligned} Z^{B}\left( \beta ,3\right) =m_{\left( 3\right) }+m_{\left( 2,1\right) }+m_{\left( 1^{3}\right) }, \end{aligned}$$

(2.6)

where we abbreviate the $m_{\left( 3\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$ to $m_{\left( 3\right) }$ and so on. For an ideal Fermi gas, a set of the occupation number $\left( a\right)$ with $a_{i}$ larger than 1 is not allowed under the Pauli exclusion principle. That is, the number of microstates $\Omega ^{ \left( a\right) } =0$ for $\left( a\right)$ with $a_{i}$ larger than 1, otherwise $\Omega ^{\left( a\right) } =1$ . Therefore, the coefficient $\Omega ^{\left( 1^{N}\right) }$ in Eq. (2.4) is 1, where $\left( 1^{N}\right)$ represents 1 apears N times, otherwise $\Omega ^{\left( \uplambda \right) _I}=0$. The canonical partition function for an ideal Fermi gas is

$$\begin{aligned} Z^{F}\left( \beta ,N\right) =m_{\left( 1^{N}\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(2.7)

For the sake of clarity, we give the explicit form of Eq. (2.7) for $N=3$:

$$\begin{aligned} Z^{F}\left( \beta ,3\right) =m_{\left( 1^{3}\right) }, \end{aligned}$$

(2.8)

For Gentile statistics, the maximum occupation number is q, thus, the number of microstates $\Omega ^{ \left( a\right) } =0$ for $\left( a\right)$ with $a_{i}$ larger than q, otherwise $\Omega ^{ \left( a\right) } =1$. Therefore, the coefficient $\Omega ^{\left( \uplambda \right) _I}$ in Eq. (2.4) with $\uplambda _{I,1}$, the largest element in the integer partition $(\uplambda )$, larger than q is 0, otherwise $\Omega ^{\left( \uplambda \right) _I}=1$. That is, the canonical partition function for an ideal Gentile gas is

$$\begin{aligned} Z^{G}\left( \beta ,N\right)= & {} \sum _{\uplambda _{I,1}\le q}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) \nonumber \\= & {} m_{\left( \uplambda \right) ^{q}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) +\sum _{\uplambda _{I,1}<q}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(2.9)

where $\left( \uplambda \right) ^{q}$ is the first integer partition with non-zero coefficient $\Omega ^{\left( \uplambda \right) _I}$ and the largest element in $\left( \uplambda \right) ^{q}$ is q. For the sake of clarity, we give the explicit form of Eq. (2.9) for $N=3$ and $q=2$:

$$\begin{aligned} Z^{G}\left( \beta ,3\right) =m_{\left( 2,1\right) }+m_{\left( 1^{3}\right) }, \end{aligned}$$

(2.10)

The result Eqs. (2.5)–(2.9) is in consistency with that in the previous work [26].

2.2 Expressing the Canonical Partition Function by the S-Function

For an ideal gas consisting of N particles, if the Hamiltonian $H_{N}$ is invariant under permutations, i.e.,

$$\begin{aligned} \left[ H,S_{N}\right] =0, \end{aligned}$$

(2.11)

where $S_{N}$ is the permutation group of order N, then the N-particle Hilbert space $V^{\otimes N}$ with V the Hilbert space of a single particle can be decomposed into Hilbert subspaces that carries the irreducible representations of $S_{N}$. For example, the symmetric and the anti-symmetric Hilbert subspaces carries the one-dimensional irreducible representation of $S_{N}$. Different Hilbert subspaces describe different particles. For example, bosons and fermions are described in the symmetric and anti-symmetric Hilbert subspaces respectively.

In this section, along this line, we give the canonical partition function of ideal gases by using the mathematical theories of the invariant matrix and the Schur-Weyl duality. We show that, the canonical partition function can be written as linear combinations of the S-function.

For an ideal gas consisting of N identical-particles, the particle should be described in one of the Hilbert subspace of $V^{\otimes N}$. The canonical partition function $Z\left( \beta ,N\right)$ is the trace of $e^{-\beta H}$ in the Hilbert subspace. We show that:

• (a) With the Schur-Weyl duality, the Hilbert subspace describing the particle can be decomposed into direct sums of “more fundamental” Hilbert subspaces that carries the inequivalent and irreducible representations of $S_{N}$. Those “more fundamental” Hilbert subspaces are labeled by integer partitions $\left( \uplambda \right)$ of N, for the sake of convenience, we denote the “more fundamental” Hilbert subspace by $V^{\left( \uplambda \right) }$. For example, for a system consisting of 3 particles, the Hilbert subspace describing the particle could be the direct sum of $V^{\left( 3\right) }$ that carries the identity representation of $S_{3}$ and $V^{\left( 2,1\right) }$ that carries a multi-dimensional irreducible representation of $S_{3}$.

• (b) With the theory of the invariant matrix, the trace of $e^{-\beta H}$ in the Hilbert subspace $V^{\left( \uplambda \right) }$ is the S-function with variables $x_{i}=e^{-\beta \varepsilon _{i}}$, $s_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1} },e^{-\beta \varepsilon _{2}},\ldots \right)$. That is,

  $$\begin{aligned} \text {Trace of } e^{-\beta H} \text {in } V^{\left( \uplambda \right) } =s_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

  (2.12)

For example, for a system consisting of 5 particles, the Hilbert space $V^{\otimes 5}$ can be decomposed into 7 subspaces, as shown in Table 1.

Table 1 An example of decomposing the Hilbert space. $V^{\left( \uplambda \right) _{I}}$ is the “more fundamental” Hilbert subspace corresponding to the Ith integer partition of N

Without loss of generality, the Hilbert subspace describing the particle can be considered as a direct sum of the “more fundamental” Hilbert subspace $V^{\left( \uplambda \right) }$ with $V^{\left( \uplambda \right) }$ occurs a given time. Therefore, the trace of $e^{-\beta H}$ in the Hilbert subspace, can be written as sums of the trace of $e^{-\beta H}$ in $V^{\left( \uplambda \right) }$, that is, the canonical partition function can be written as linear combinations of the S-function.

The canonical partition function of an ideal gas consisting of N identical-particles can be written as a linear combination of S-function, that is,

$$\begin{aligned} Z\left( \beta ,N\right) =\sum _{I}^{P\left( N\right) }C^{\left( \uplambda \right) _I}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(2.13)

where $C^{\left( \uplambda \right) _I}\$ is an non-negative integer and represents the times $V^{\left( \uplambda \right) _{I} }$ occurs in the Hilbert subspace that describes the system.

The rigorous calculation of Eq. (2.13) is given later in this section.

The general formula of the canonical partition function of ideal gases, Eq. (2.13), shows that the canonical partition function of ideal gases can be written as linear combinations of the S-function and is one the main result in the present paper. We will show that the coefficient of the S-function, $C^{\left( \uplambda \right) _I}$, in Eq. (2.13) carries important information such as the permutation phase of the wave function and the commutation relation between the Hamiltonian and the permutation group in Sect. 3.

Examples: Bose statistics, Fermi statistics, and parastatistics. For Bose statistics, the particle is described in the symmetric Hilbert subspace $V^{\left( \uplambda \right) }=V^{\left( N\right) }$, thus, the canonical partition function is

$$\begin{aligned} Z^{B}\left( \beta ,N\right) =s_{\left( N\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(2.14)

For Fermi statistics, the particle is described in the anti-symmetric Hilbert subspace $V^{\left( \uplambda \right) }=V^{\left( 1^{N}\right) }$, thus, the canonical partition function is

$$\begin{aligned} Z^{F}\left( \beta ,N\right) =s_{\left( 1^{N}\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(2.15)

For para-Bose with the parameters q, the particle is described in the Hilbert subspaces where $V^{\left( \uplambda \right) _I}$ with the number of elements in the integer partition $\left( \uplambda \right)$ smaller than q occurs once [4, 19, 27]. For para-Fermi statistics with the parameters q, the particle is described in the Hilbert subspaces where $V^{\left( \uplambda \right) _I}$ with the maximum element in the integer partition $\left( \uplambda \right)$ smaller than q occurs once [4, 19, 27]. Therefore, the canonical partition function is

$$\begin{aligned} Z_{q}^{PB}\left( \beta ,N\right)&=\sum _{l_{\left( \uplambda \right) _{I} }\le q}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1} },e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(2.16)

$$\begin{aligned} Z_{q}^{PF}\left( \beta ,N\right)&=\sum _{\uplambda _{I,1}\le q}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(2.17)

where $l_{\left( \uplambda \right) }$ is the number of elements in the integer partition $\left( \uplambda \right)$ and $\uplambda _{I,1}$ is the largest element in $\left( \uplambda \right) _{I}$. Equations (2.14) and (2.15) are in consistency with that in the previous work [26]. Equations (2.16) and (2.17) are in consistency with the result in Ref. [28].

Here is the detail of the calculation of Eq. (2.13).

A brief review on the Schur-Weyl duality. Let V be a linear space of the dimension m. Let $g\in GL\left( V\right)$ be a linear operator on V. The action of g on a vector $e_{i_{1}}\otimes e_{i_{2}}\otimes \ldots \otimes e_{i_{N}}$ in $V^{\otimes N}$ is

$$\begin{aligned} g\left( e_{i_{1}}\otimes e_{i_{2}}\otimes \ldots \otimes e_{i_{N}}\right) \equiv ge_{i_{1}}\otimes ge_{i_{2}}\otimes \ldots \otimes ge_{i_{N}}, \end{aligned}$$

(2.18)

where, $\left\{ e_{1},e_{2},\ldots \right\}$ is a basis in V. For $\sigma \in S_{N}$, the action of $\sigma$ on the vector $e_{i_{1}}\otimes e_{i_{2} }\otimes \ldots \otimes e_{i_{N}}$ is

$$\begin{aligned} \sigma \left( e_{i_{1}}\otimes e_{i_{2}}\otimes \ldots \otimes e_{i_{N}}\right) \equiv e_{\sigma _{i_{1}}}\otimes e_{\sigma _{i_{2}}}\otimes \ldots \otimes e_{\sigma _{i_{N}}}. \end{aligned}$$

(2.19)

Equations (2.18) and (2.19) imply that the operator g commute with $\sigma$ on $V^{\otimes N}$. For $m\ge N$, the space $V^{\otimes N}$ can be decomposed into a direct sum of the subspace $V^{\left( \uplambda \right) _I}$ [23, 29]. Each integer partition $\left( \uplambda \right) _{I}$ of N corresponds to a subspace $V^{\left( \uplambda \right) _I}$ [23, 29] and the number of $V^{\left( \uplambda \right) _I}$ is $P\left( N\right)$, i.e.,

$$\begin{aligned} V^{\otimes N}= {\displaystyle \bigoplus \limits _{I=1}^{P\left( N\right) }} V^{\left( \uplambda \right) _I}. \end{aligned}$$

(2.20)

The subspace $V^{\left( \uplambda \right) _I}$ carries irreducible representations for $S_{N}$ and $GL\left( V\right)$ with the dimension $f_{I}$ and $R_{I}$ respectively, where

$$\begin{aligned} f_{I}=N! {\displaystyle \prod \limits _{i=1,i<j}} \left( \uplambda _{I,i}-\uplambda _{I,j}-i+j\right) {\displaystyle \prod \limits _{i=1}} \left[ \left( l_{\left( \uplambda \right) }+\uplambda _{I,i}-i\right) !\right] ^{-1}, \end{aligned}$$

(2.21)

and

$$\begin{aligned} R_{I}= {\displaystyle \prod \limits _{i<j}^{m}} \left( a_{I,i}-a_{I,j}+j-i\right) \left( j-i\right) ^{-1} \end{aligned}$$

(2.22)

with $a_{I,1}=\uplambda _{I,1}$, $a_{I,2}=\uplambda _{I,2}$, $\ldots$, $a_{I,l_{\left( \uplambda \right) }}=\uplambda _{I,l_{\left( \uplambda \right) }}$, $a_{I,l_{\left( \uplambda \right) }+1}=0$,$\ldots$, and $a_{I,m}=0$ [23, 29]. The dimension of $V^{I}$ is [23, 29]

$$\begin{aligned} \text {dim}\left( V^{\left( \uplambda \right) _I}\right) =f_{I}R_{I}. \end{aligned}$$

(2.23)

For $S_{N}$, the inequivalent and irreducible representation with the dimension $f_{I}$ occurs $R_{I}$ times in $V^{\left( \uplambda \right) _I}$ [23, 29]. For $GL\left( V\right)$, the inequivalent and irreducible representation of the dimension $R_{I}$ occurs $f_{I}$ times in $V^{\left( \uplambda \right) _I}$ [23, 29]. It can be verified that

$$\begin{aligned} \text {dim}\left( V^{\otimes N}\right) =m^{N}=\sum _{I=1}^{P\left( N\right) } \text {dim}\left( V^{\left( \uplambda \right) _I}\right) =\sum _{I=1}^{P\left( N\right) }R_{I}f_{I}. \end{aligned}$$

(2.24)

A brief review on the mathematical theory of the invariant matrix. Let G be an m-dimensional matrix group. Let A be an m-dimensional matrix in G. Let $T\left( A\right)$ be a matrix whose elements are polynomials in the elements of A. $T\left( A\right)$ is an invariant matrix [22] if

$$\begin{aligned} T\left( A\right) T\left( B\right) =T\left( AB\right) , \end{aligned}$$

(2.25)

where B is also an m-dimensional matrix in G. The invariant matrix gives a representation of the group G. If T is reducible, then for any A in G, $T\left( A\right)$ can be diagonalized in the same way and the matrix in diagonal is a new invariant matrix of G [22]. The N times direct product of G, $G^{\otimes N}$, is an invariant matrix [22]. The $G^{\otimes N}$ can be decomposed into $P\left( N\right)$ irreducible invariant matrices. An integer partition $\left( \uplambda \right) _{I}$ corresponds to an irreducible invariant matrix, denoted by $T^{I}\left( G\right)$. For A in G, the trace of $T^{I}\left( A\right)$ is [22]

$$\begin{aligned} tr\left[ T^{I}\left( A\right) \right] =s_{\left( \uplambda \right) _{I} }\left( a_{1},a_{2},\ldots \right) , \end{aligned}$$

(2.26)

where $a_{i}$ is the ith eigenvalue of A.

The direct sum decomposition of the N-particle Hilbert space $V^{\otimes N}$. With the Schur-Weyl duality, the Hilbert space of a N-particle system $V^{\otimes N}$ can be decomposed into a direct sum of subspaces $V^{\left( \uplambda \right) _I}$. The number of the subspace is $P\left( N\right)$ and each integer partition $\left( \uplambda \right) _{I}$ of N corresponds to a subspace $V^{\left( \uplambda \right) _I}$. The dimension of the subspace $V^{\left( \uplambda \right) _I}$ is $R_{I}f_{I}$. The space $V^{\left( \uplambda \right) _I}$ gives $f_{I}$ equivalent and irreducible representations with the dimension $R_{I}$ for the Hamiltonian and $R_{I}$ equivalent and irreducible representations with the dimension $f_{I}$ for $S_{N}$.

An example of decomposing the Hilbert space $V^{\otimes N}$. For the sake of clarity, we give an example of the decomposition of the $V^{\otimes N}$. Let the dimension of the Hilbert space of a single particle V be 6. For a system consists of 5 particles, the Hilbert space $V^{\otimes 5}$ is a 7776-dimensional space. It can be decomposed into 7 subspaces, as shown in Table 2: the subspace $V^{\left( 5\right) }$ corresponds to the integer partition $\left( 5\right)$ and the dimension of $V^{\left( 5\right) }$ is 252. It gives a one-dimensional representation of $S_{N}$, and a 252-dimensional representation of the Hamiltonian, and so on.

Table 2 An example of decomposing the Hilbert space. A system consists of 5 particles with the dimension of the Hilbert space of a single particle V being 6

The trace of the operator $e^{-\beta H}$ in the subspace. Let H be the Hamiltonian of a single particle and V be the Hilbert space of a single particle. One can give the matrix expression of the operator $e^{-\beta H}$ on V

$$\begin{aligned} e^{-\beta H}=\sum _{i=1}e^{-\beta \varepsilon _{i}}|\phi _{i}\rangle \langle \phi _{i}|, \end{aligned}$$

(2.27)

where $|\phi _{i}\rangle$ is the eigenfunction of the Hamiltonian H and $\varepsilon _{i}$ is the corresponding eigenvalue. $e^{-\beta H_{N}}=\left( e^{-\beta H}\right) ^{\otimes N}$ is an operator on $V^{\otimes N}$, where $H_{N}= {\displaystyle \bigoplus \nolimits _{i=1}^{N}} H_{i}$ is the Hamiltonian of an N-identical-particle gas system. Since $e^{-\beta H_{N}}$ is an invariant matrix of $e^{-\beta H}$, it can be decomposed into $P\left( N\right)$ irreducible invariant matrices. Each irreducible invariant matrix, denoted by $D^{I}\left( e^{-\beta H}\right)$, corresponds to an integer partition $\left( \uplambda \right) _{I}$ of N. The trace of $D^{I}\left( e^{-\beta H}\right)$ is

$$\begin{aligned} tr\left[ D^{I}\left( e^{-\beta H}\right) \right] =s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(2.28)

By the mathematical theory of the Schur-Weyl duality, we recognize that Eq. (2.28) is the trace of $e^{-\beta H_{N}}$ in the subspace $V^{\prime \left( \uplambda \right) _{I}}$ which carries the inequivalent and irreducible representation corresponding to the integer partition $\left( \uplambda \right) _{I}$. That is, for a complete basis $\left| \Phi \right\rangle$ in $V^{\prime \left( \uplambda \right) _{I}}$, one has

$$\begin{aligned} \sum \left\langle \Phi \right| e^{-\beta H_{N}}\left| \Phi \right\rangle =s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}} ,e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(2.29)

According to the Schur-Weyl duality, the inequivalent and irreducible representations $V^{\prime \left( \uplambda \right) _{I}}$ occurs $f_{I}$ times in $V^{\left( \uplambda \right) _I}$. That is, for a complete basis $\left| \Psi \right\rangle$ in $V^{\left( \uplambda \right) _I}$, the equation

$$\begin{aligned} \sum \left\langle \Psi \right| e^{-\beta H_{N}}\left| \Psi \right\rangle =f_{I}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1} },e^{-\beta \varepsilon _{2}},\ldots \right) \end{aligned}$$

(2.30)

holds. In Eq. (2.30), the coefficient $f_{I}$ can be canceled by setting

$$\begin{aligned} \left| \Psi ^{\prime }\right\rangle =\frac{1}{\sqrt{f_{I}}}\left| \Psi \right\rangle . \end{aligned}$$

(2.31)

Thus, we make no distinguish between the subspace $V^{\prime \left( \uplambda \right) _{I}}$ and $V^{\left( \uplambda \right) _I}$ in the rest discussion of the present paper.

The calculation of Eq. (2.13). An identical-particle system is described in a Hilbert subspace D. The space D can be decomposed into subspace $V^{\left( \uplambda \right) _I}$ that carries the equivalent and irreducible representation of $S_{N}$ [30], i.e.,

$$\begin{aligned} D= {\displaystyle \bigoplus \limits _{I=1}^{P\left( N\right) }} \left( V^{\left( \uplambda \right) _I}\right) ^{\oplus C^{\left( \uplambda \right) _I}}, \end{aligned}$$

(2.32)

where $C^{\left( \uplambda \right) _I}$ are nonnegative integers representing the times of $V^{\left( \uplambda \right) _I}$ occuring in D. By the definition of the canonical partition function, $Z\left( \beta ,N\right) =tr\left[ D\left( e^{-\beta H}\right) \right]$, and Eq. (2.28), we give the canonical partition function of an N-identical-particle system:

$$\begin{aligned} Z\left( \beta ,N\right)&=tr\left[ D\left( e^{-\beta H}\right) \right] \nonumber \\&=\sum _{I=1}^{P\left( N\right) }C^{\left( \uplambda \right) _I}tr\left[ D^{I}\left( e^{-\beta H}\right) \right] \nonumber \\&=\sum _{I=1}^{P\left( N\right) }C^{\left( \uplambda \right) _I}s_{\left( \uplambda \right) _{I} }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(2.33)

where $D\left( e^{-\beta H}\right)$ is the representation of $e^{-\beta H}$ on D. Therefore, Eq. (2.33) yields Eq. (2.13).

3 The Coefficient in the Canonical Partition Function

In this section, we show that for an ideal gas, once the canonical partition function is written as linear combinations of the M-function, Eq. (2.4), or the S-function, Eq. (2.13), the coefficient of the M-function and the S-function, $\Omega ^{\left( \uplambda \right) _I}$ and $C^{\left( \uplambda \right) _I}$ , contain information of the maximum occupation number, the indistinguishability of particles, the permutation phase of the wave function, and the commutation relation between the Hamiltonian and the permutation group.

3.1 The Maximum Occupation Number

In this section, we suggest a method to obtain the maximum occupation number of the system from the canonical partition function.

For an ideal gas, if the canonical partition function is written as linear combinations of the M-function, Eq. (2.4), the first non-zero coefficient of the M-function , $\Omega ^{\left( \uplambda \right) _I}$, gives the maximum occupation number $q=\uplambda _{I,1}$ with $\uplambda _{I,1}$ the largest element in $\left( \uplambda \right) _{I}$.

The integer partition of N, $\uplambda _{I}$, is arranged in a prescribed order: for $I<J$, $\uplambda _{I,1}\ge \uplambda _{J,1}$. If the coefficient $\Omega ^{\left( \uplambda \right) _I}$ is not zero, the microstate with the maximum occupation number $\uplambda _{I,1}$ is allowed. Therefore, the first non-zero coefficient $\Omega ^{\left( \uplambda \right) _I}$ will give the largest $\uplambda _{I,1}$ as well as the maximum occupation number q.

Examples. The canonical partition function of a Gentile gas with particles number $N=4$ and the maximum occupation number $q=2$ is

$$\begin{aligned} Z^{G}\left( \beta ,4\right) =m_{\left( 2^{2}\right) }+m_{\left( 2,1^{2}\right) }+m_{\left( 1^{4}\right) }, \end{aligned}$$

(3.1)

where the first non-zero coefficient is $\Omega ^{\left( \uplambda \right) _{3}}=\Omega ^{\left( 2^{2}\right) }=1$ ($\Omega ^{\left( \uplambda \right) _{1}}=\Omega ^{\left( 4\right) }=0$, $\Omega ^{\left( \uplambda \right) _{2}}=\Omega ^{\left( 3,1\right) }=0$). Therefore, the maximum occupation number is 2. Moreover, for an ideal gas with the maximum occupation number q, the canonical partition function can be written as

$$\begin{aligned} Z\left( \beta ,N\right) =m_{\left( \uplambda \right) ^{q}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) +\sum _{\uplambda _{I,1}\le q}^{P\left( N\right) }\Omega ^{\prime I}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(3.2)

where $\left( \uplambda \right) ^{q}$ denotes the first non-zero-coefficient integer partition with $\uplambda _{1}=q$.

3.2 The Indistinguishability of Particles

In this section, we suggest a method to obtain the indistinguishability of particles from the canonical partition function.

If the particle is indistinguishable, then exchanging two particles will not lead to any observable effect. One of the necessary condition is that exchanging two particles that occupy different quantum states will not lead to a new microstate.

For an ideal gas consisting of N particles, if the particles are indistinguishable, the coefficient of the M-function $\Omega ^{\left( \uplambda \right) _I}$ in the canonical partition function, Eq. (2.4), should be either 1 or 0. That is, if any coefficients of the M-function $\Omega ^{\left( \uplambda \right) _I}$ in the canonical partition function, Eq (2.4), is larger than 1 (neither 1 nor 0 ), the particles are not indistinguishable.

For indistinguishable particles, the number of microstates $\Omega ^{ \left( a\right) }$ is 1 when giving a specified set of the occupation number $\left( a\right)$. It is because exchanging two particles that occupy different quantum states will not lead to a new microstate. Therefore, the coefficient of the M-function, $\Omega ^{\left( \uplambda \right) _I}$, should be either 1 or 0. For example, if the occupation number violates the Pauli exclusion principle, then the coefficient of the M-function, $\Omega ^{\left( \uplambda \right) _I}$, is 0. In other words, if any coefficients of the M-function, $\Omega ^{\left( \uplambda \right) _I}$, is larger than 1 (neither 1 nor 0), the particles in the system are not indistinguishable.

Examples. The canonical partition of a three-para-fermions system with the parameter $q=2$, of which the detail of the calculation will be given in the following section, is

$$\begin{aligned} Z_{2}^{PF}\left( \beta ,3\right) =m_{\left( 2,1\right) }+3m_{\left( 1^{3}\right) }. \end{aligned}$$

(3.3)

Because the coefficient $\Omega ^{\left( \uplambda \right) _I}$ in Eq. (3.3) is neither 1 nor 0, for example, $\Omega ^{\left( 1^{3}\right) }=3$, the para-fermions is not indistinguishable. The indistinguishability of para particles will be discussed in Sect. 5.1.

To be noticed that the specified value of the coefficient of the M-function evaluates the indistinguishability of particles. For example, the canonical partition of a three-para-fermions system with parameter $q=2$, is given in Eq. (3.3). For a three-Gentile particles system with the maximum occupation number $q=2$, the canonical partition function is

$$\begin{aligned} Z_{2}^{G}\left( \beta ,3\right) =m_{\left( 2,1\right) }+m_{\left( 1^{3}\right) }. \end{aligned}$$

(3.4)

From Eqs. (3.3) and (3.4) one can see that, although they share the same maximum occupation number, 2, they are different statistics because they have different coefficients of the M-function. For Gentile particles, exchanging two particles leads to no new microstates. However, for para particles, exchanging two particles might leads to new microstates. Thus para particles are not indistinguishable. The maximum occupation number together with the specified value of the coefficient of the M-function, or the indistinguishability of particles, determine a kind of statistics.

3.3 The Permutation Phase of the Wave Function and Hilbert Subspaces

In this section, we suggest a method to obtain the permutation phase of the wave function and Hilbert subspaces from the canonical partition function.

For an ideal gas, if the canonical partition function is written as linear combinations of the S-function, Eq. (2.13), the coefficient of the S-function $C^{\left( \uplambda \right) _I}$ directly gives both the structure the permutation phase of the wave function and the Hilbert subspaces that describes the particle:

• (a) In the Hilbert subspaces D that describes the particle, the Hilbert subspace $V^{\left( \uplambda \right) _I}$ that carries the inquivalent and irreducible representation of $S_{N}$ occurs $C^{\left( \uplambda \right) _I}$ times. That is

  $$\begin{aligned} D= {\displaystyle \bigoplus \limits _{I=1}^{P\left( N\right) }} \left( V^{\left( \uplambda \right) _I}\right) ^{\oplus C^{\left( \uplambda \right) _I}}. \end{aligned}$$

  (3.5)

• (b) The wave function $|\Phi \rangle$ of the system spans D, therefore, after exchanging two particles, $\sigma _{ij} |\Phi \rangle$ satisfies

  $$\begin{aligned} \sigma _{ij}|\Phi \rangle =D\left( \sigma _{ij}\right) |\Phi \rangle , \end{aligned}$$

  (3.6)

  where $\sigma _{ij}|\Phi \rangle$ represents exchanging the ith particle and the jth particle in the wave function $|\Phi \rangle$. $D\left( \sigma _{ij}\right)$ here is the permutation phase,

  $$\begin{aligned} D\left( \sigma _{ij}\right) = {\displaystyle \bigoplus \limits _{I=1}^{P\left( N\right) }} \left[ D^{I}\left( \sigma _{ij}\right) \right] ^{\oplus C^{\left( \uplambda \right) _I}}, \end{aligned}$$

  (3.7)

  where $D^{I}$ is the inequivalent and irreducible representation of $S_{N}$ corresponding to the integer partition $\left( \uplambda \right) _{I}$, and $D^{I}$ occurs $C^{\left( \uplambda \right) _I}$ times.

Examples. For a three-para-fermions system with parameter $q=2$, the canonical partition is given in Eq. (3.3). The Hilbert subspace describing the system is

$$\begin{aligned} D=V^{\left( 2,1\right) }+V^{\left( 1^{3}\right) }. \end{aligned}$$

(3.8)

The permutation phase of the wave function is

$$\begin{aligned} D\left( \sigma _{ij}\right) =\left[ \begin{array}{cc} D^{\left( 2,1\right) }\left( \sigma _{ij}\right) &{} 0 \\ 0 &{} D^{\left( 1^{3}\right) }\left( \sigma _{ij}\right) \end{array} \right] , \end{aligned}$$

(3.9)

where $D^{\left( 1^{3}\right) }\left( \sigma _{ij}\right)$ is the anti-symmetric representation of $S_{3}$ and $D^{\left( 2,1\right) }\left( \sigma _{ij}\right)$ is the inequivalent and irreducible representation of $S_{3}$ labeled by $\left( 2,1\right)$.

3.4 The Commutation Relation Between the Hamiltonian and the Permutation Group

In this section, we suggest a method to obtain the commutation relation between the system’s Hamiltonian and the permutation group from the canonical partition function.

For an ideal gas, if the Hamiltonian is invariant under permutations, i.e. $\left[ H,S_{N}\right] =0$ then the canonical partition function can be written as linear combinations of the S-function, Eq. (2.13), with non-negative coefficients of the S-function $C^{\left( \uplambda \right) _I}$, which is proved in Sect. 2.2. That is, if any coefficients of the S-function $C^{\left( \uplambda \right) _I}$ in the canonical partition function, Eq. (2.13), are negative, the Hamiltonian is noninvariant under permutations, i.e. $\left[ H,S_{N}\right] \ne 0$. which means that the particle is not indistinguishable.

Examples. For a three-Gentile particles system with the maximum occupation number $q=2$, the canonical partition function, of which the detail of the calculation will be given in the following section, is

$$\begin{aligned} Z_{2}^{G}\left( \beta ,3\right) =s_{\left( 2,1\right) }-s_{\left( 1^{3}\right) }. \end{aligned}$$

(3.10)

Because the coefficient $C^{\left( \uplambda \right) _I}$ in Eq. (3.10) is not non-negative, $C^{\left( 1^{3}\right) }=-1$. The Hamiltonian of ideal Gentile gases is noninvariant under permutations. The indistinguishability of Gentile particles will be discussed in Sect. 5.3.

4 The Connection Between the Permutation Phase of the Wave Function and the Maximum Occupation Number

In this section, by using the relation between the M-function and the S-function, we reveal the connection between the permutation phase of the wave function and the maximum occupation number. Moreover, by using this relation, we show that, bosons and fermions are the only two kinds of indistinguishable particles.

4.1 The Permutation Phase and the Maximum Occupation Number

The S-function can be represented as a linear combination of the M-function [22, 24],

$$\begin{aligned} s_{\left( \uplambda \right) _{K}}\left( x_{1},x_{2},\ldots \right) =\sum _{I=1}^{P\left( N\right) }k_{K}^{I}m_{\left( \uplambda \right) _{I}}\left( x_{1},x_{2},\ldots \right) , \end{aligned}$$

(4.1)

where $k_{K}^{I}$ is the Kostka number, an important constant in the number theory.

In this section, we reveal the connection between the permutation phase of the wave function and the maximum occupation number.

For an ideal gas consisting of N particles, the canonical partition function can be written in terms of the M-function, Eq. (2.4), and the S-function, Eq. (2.13), respectively. The maximum occupation is embedded in the coefficient of the M-function $\Omega ^{\left( \uplambda \right) _I}$ and the permutation phase is embedded in the coefficient of the S-function $C^{\left( \uplambda \right) _J}$, the coefficient is connected by

$$\begin{aligned} \Omega ^{\left( \uplambda \right) _I}=\sum _{J}^{P\left( N\right) }k_{J}^{I}C^{\left( \uplambda \right) _J} \end{aligned}$$

(4.2)

where $k_{J}^{I}$ is the Kostka number.

Substituting Eq. (4.1) into Eq. (2.13) and equaling Eqs. (2.4) and (2.13) give Eq. (4.2).

Examples: parastatistics and Gentile statistics. The Kostka number $k_{K}^{I}$ is labeled by the Ith and the Kth integer partition of N. For $N=3$, the Kostka number $k_{K}^{J}$ is $k_{\left( 3\right) }^{\left( 3\right) }=k_{1}^{1}=1$, $k_{\left( 2,1\right) }^{\left( 3\right) }=k_{2}^{1}=0$, $k_{\left( 1^{3}\right) }^{\left( 3\right) }=k_{3}^{1}=0$, $k_{\left( 3\right) }^{\left( 2,1\right) } =k_{1}^{2}=1$, $k_{\left( 2,1\right) }^{\left( 2,1\right) }=k_{2}^{2}=1$, $k_{\left( 1^{3}\right) }^{\left( 2,1\right) }=k_{3}^{2}=0$, $k_{\left( 3\right) }^{\left( 1^{3}\right) }=k_{1}^{3}=1$, $k_{\left( 2,1\right) }^{\left( 1^{3}\right) }=k_{2}^{3}=2$, and $k_{\left( 1^{3}\right) }^{\left( 1^{3}\right) }=k_{3}^{3}=1$ [22, 24]. For clarity , we rewrite the Kostka number in a matrix form:

$$\begin{aligned} k=\left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) , \end{aligned}$$

(4.3)

where we take the upper index of $k_{K}^{J}$ as the row index and the lower index as the column index.

For a three-para-fermions system with the parameter $q=2$, the canonical partition function expressed in S-function is,

$$\begin{aligned} Z_{2}^{PF}\left( \beta ,3\right) =s_{\left( 2,1\right) }+s_{\left( 1^{3}\right) }. \end{aligned}$$

(4.4)

that is $C=\left( 0,1,1\right) ^{T}$. By applying the transform Eq. (4.2) with the Kostka number given in Eq. (4.3), we have

$$\begin{aligned} \left( \begin{array} {c} 0\\ 1\\ 3 \end{array} \right) =\left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c}0\\ 1\\ 1 \end{array} \right) , \end{aligned}$$

i.e., $\Omega =\left( 0,1,3\right) ^{T}$, that is

$$\begin{aligned} Z_{2}^{PF}\left( \beta ,3\right) =m_{\left( 2,1\right) }+3m_{\left( 1^{3}\right) }. \end{aligned}$$

(4.5)

From the special case, Eqs. (5.7) and (4.5), one can see that, firstly, the maximum occupation number for para-fermions with the parameter $q=2$ is 2 and there is no limitation on the maximum occupation number for para-bosons. Secondly, the para particles are not indistinguishable.

For a system consisting of three Gentile particles and the maximum occupation number $q=2$, the canonical partition function expressed in term of the M-function is,

$$\begin{aligned} Z_{2}^{G}\left( \beta ,3\right) =m_{\left( 2,1\right) }+m_{\left( 1^{3}\right) }. \end{aligned}$$

(4.6)

That is $\Omega =\left( 0,1,1\right) ^{T}$. By applying the transform Eq. (4.2) with the Kostka number given in Eq. (4.3), we have $C=\left( 0,1,-1\right) ^{T}$, that is

$$\begin{aligned} Z_{2}^{G}\left( \beta ,3\right) =s_{\left( 2,1\right) }-s_{\left( 1^{3}\right) }. \end{aligned}$$

(4.7)

From the special case, Eq. (4.7), one can see that, the Hamiltonian of the Gentile system is not invariant under permutations. For a three-para-bosons system with the parameter $q=2$, the canonical partition function expressed in terms of the S-function is,

$$\begin{aligned} Z_{2}^{PB}\left( \beta ,3\right) =s_{\left( 3\right) }+s_{\left( 2,1\right) }. \end{aligned}$$

(4.8)

That is $C^{\left( \uplambda \right) _{1}}=C^{\left( 3\right) }=1$, $C^{\left( \uplambda \right) _{2}}=C^{\left( 2,1\right) }=1$, $C^{\left( \uplambda \right) _{3}}=C^{\left( 1^{3}\right) }=0$. For the sake of convenience, we denote it by $C=\left( 1,1,0\right) ^{T}$. By applying the transform Eq. (4.2) with the Kostka number given in Eq. (7.9), we have $\Omega =\left( 1,2,3\right) ^{T}$, that is

$$\begin{aligned} Z_{2}^{PB}\left( \beta ,3\right) =m_{\left( 3\right) }+2m_{\left( 2,1\right) }+3m_{\left( 1^{3}\right) }. \end{aligned}$$

(4.9)

4.2 Bosons and Fermions: Indistinguishable Particles

If the particle is indistinguishable, on the one hand exchanging two particles that occupy different quantum states will not lead to a new microstate. On the other hand, the Hamiltonian should be invariant under permutations. It shows in Sects. 3.2 and 3.4 that, if the particle is indistinguishable, then, the coefficient of the M-function $\Omega ^{\left( \uplambda \right) _{I}}$ in Eq. (5.46) should be either 1 or 0 (exchanging two particles that occupy different quantum states will not lead to a new microstate) and the coefficient of the S-function $C^{\left( \uplambda \right) _{J}}$ in Eq. (5.46) should be non-negative (the Hamiltonian is invariant under permutation).

Therefore, the solution to Eq. (4.2) with $\Omega ^{\left( \uplambda \right) _{I}}$ either 1 or 0 and $C^{\left( \uplambda \right) _{J}}$ non-negative gives a kind of indistinguishable particles. The number of such solutions is the number of different kinds of indistinguishable particles allowed.

Although to compute the Kostka number at large N is difficult [31], the Kostka number $k_{J}^{I}$ has following properties: for $J>I$, $k_{J}^{I}=0$. For $I=J$, $k_{J}^{I}=1$. for $J=1$, $k_{J}^{I}=1$. otherwise $k_{J}^{I}\ge 1$ [22, 24].

By using the property of the Kostka number, we give a conjecture that there are only two solutions to Eq. (4.2) that with $\Omega ^{\left( \uplambda \right) _{I}}$ either 1 or 0 and $C^{\left( \uplambda \right) _{J}}$ non-negative. They are

$$\begin{aligned} C^{\left( \uplambda \right) _{I}}=\left\{ \begin{array}{c} \!\!1\text {, if }I=1\\ 0\text {, otherwise} \end{array} \right. \text { and }\Omega ^{\left( \uplambda \right) _{I}}=\Omega ^{\left( \uplambda \right) _{J}}=1, \end{aligned}$$

(4.10)

which yields bosons with the canonical partition function $Z^{B}\left( \beta ,N\right) = s_{\left( N\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$ $=\sum _{I=1}^{P\left( N\right) }m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$, and

$$\begin{aligned} C^{\left( \uplambda \right) _{I}}=\left\{ \begin{array}{l} 1\text {, if }I=P\left( N\right) \\ 0\text {, otherwise} \end{array} \right. \text { and }\quad \Omega ^{\left( \uplambda \right) _{I}}=\left\{ \begin{array}{l} 1\text {, if }I=P\left( N\right) \\ 0\text {, otherwise} \end{array} \right. , \end{aligned}$$

(4.11)

which yields fermions with the canonical partition function $Z^{F}\left( \beta ,N\right) =s_{\left( 1^{N}\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2} },\ldots \right)$ $=m_{\left( 1^{N}\right) }\left( e^{-\beta \varepsilon _{1} },e^{-\beta \varepsilon _{2}},\ldots \right)$. Thus, bosons and fermions are the only indistinguishable particles.

For example, for the system consisting of three bosons the coefficient of the M-function reads: $\Omega ^{\left( \uplambda \right) _{1}}=\Omega ^{\left( 3\right) }=1$, $\Omega ^{\left( \uplambda \right) _{2}}=\Omega ^{\left( 2,1\right) }=1$, and $\Omega ^{\left( \uplambda \right) _{3}}=\Omega ^{\left( 1^{3}\right) }=1$. The coefficient of the S-function reads: $C^{\left( \uplambda \right) _{1} }=C^{\left( 3\right) }=1$, $C^{\left( \uplambda \right) _{2}}=C^{\left( 2,1\right) }=0$, and $C^{\left( \uplambda \right) _{2}}=C^{\left( 2,1\right) }=0$. Eq. (4.2) now is

$$\begin{aligned} \left( \begin{array}{c} 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 0\\ 0 \end{array} \right) . \end{aligned}$$

For the system consisting of three fermions the coefficient of the M-function reads: $\Omega ^{\left( \uplambda \right) _{1}}=\Omega ^{\left( 3\right) }=0$, $\Omega ^{\left( \uplambda \right) _{2}}=\Omega ^{\left( 2,1\right) }=0$, and $\Omega ^{\left( \uplambda \right) _{3}}=\Omega ^{\left( 1^{3}\right) }=1$. The coefficient of the S-function reads: $C^{\left( \uplambda \right) _{1}}=C^{\left( 3\right) }=0$, $C^{\left( \uplambda \right) _{2}}=C^{\left( 2,1\right) }=0$, and $C^{\left( \uplambda \right) _{2}}=C^{\left( 2,1\right) }=1$. Eq. (4.2) now is

$$\begin{aligned} \left( \begin{array}{c} 0\\ 0\\ 1 \end{array} \right) =\left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 0\\ 0\\ 1 \end{array} \right) . \end{aligned}$$

Other solutions, either have the coefficient of the M-function larger than 1 (neither 1 nor 0), or have the coefficient of the S-function negative. That is, for other particles obeying generalized statistics, exchanging two particles lead to either a new microstate or the result that the Hamiltonian is noninvariant under permutations. For example, the coefficient of the M-function reads: $\Omega ^{\left( \uplambda \right) _{1}}=\Omega ^{\left( 3\right) }=0$, $\Omega ^{\left( \uplambda \right) _{2}}=\Omega ^{\left( 2,1\right) }=1$, and $\Omega ^{\left( \uplambda \right) _{3}}=\Omega ^{\left( 1^{3}\right) }=1$. Then, by Eq. (4.2), the coefficient of the S-function is given as: $C^{\left( \uplambda \right) _{1}}=C^{\left( 3\right) }=0$, $C^{\left( \uplambda \right) _{2}}=C^{\left( 2,1\right) }=1$, and $C^{\left( \uplambda \right) _{2} }=C^{\left( 2,1\right) }=-1$. Because the coefficient $C^{\left( 2,1\right) }=-1$, thus the Hamiltonian of particles is noninvariant under permutations, and particles are not indistinguishable.

5 A Unified Framework

Although, fundamental particles such as electrons and photons in real world are either bosons or fermions, there are still elementary excitations in many-body systems may obey generalized statistics. For example, Gentile statistics describes the spin wave [6], particles obeying para-statistics are candidates of dark matter and/or dark energy [14]. Thus, a deeper understanding of such generalized statistics is needed. For example, what is the permutation phase of Gentile particles and what is the connection between those particles, or do they share some commonalities.

It shows in Sects. 3, 4, and 5 that by expressing the canonical partition function as linear combinations of the M-function or the S-function, one can obtain the maximum occupation number, the indistinguishability of the particle, the permutation phase of the wave function, and the commutation relation between the Hamiltonian and the permutation group by analyzing the combination coefficients.

In this section, by using the result obtained in Sect. 3, 4, and 5, we describe a series of generalized statistics in a unified framework. Such generalized statistics are parastatistics [7, 8], the intermediate statistics or Gentile statistics [11, 12], Gentileonic statistics [10], and immannons [9]. We discuss the missing property of such generalized particles, e.g., the permutation phase of Gentile particles and the occupation number of para-particles. We show that particles obeying those kinds of statistics are not indistinguishable and usually have a maximum occupation number. We also show that the Gentileonic statistics and immannons are essentially the same particles. Especially, we propose three new generalized statistics, which seem to be the missing pieces in the puzzle.

The Hilbert subspace helps to illustrate the difference between generalized statistics intuitively. For example, the Hilbert subspace describing Bose statistics is the symmetric subspace, as shown in Fig. 1. The Hilbert subspace describing Fermi statistics is the anti-symmetric subspace, as shown in Fig. 2. Therefore, we also depict the Hilbert subspace that describs those particles.

Fig. 1

The Hilbert subspace describing Bose–Einstein statistics. Each cube represents a subspace corresponding to an integer partition

Fig. 2

The Hilbert subspace describing Fermi-Dirac statistics

5.1 Parastatistics

Parastatitics is proposed by H. Green as a generalization of Bose and Fermi statistics in 1953 [7, 17]. In Green’s generalization, a trilinear relation of the algebra of creation and annihilation operators is proposed [28, 32]. One already knows that parastatitics corresponding to the higher dimensional representation of the permutation group [4]. For instance, Okayama (1952) [19] suggests that all irreducible representations associated with Young diagrams of at most p columns yield parastatitics [27]. Canonical partition functions of para-Bose and para-Fermi statistics with the parameter q are [28]

$$\begin{aligned} Z_{q}^{PB}\left( \beta ,N\right)&=\sum _{l_{\left( \uplambda \right) }\le q}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}} ,e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.1)

$$\begin{aligned} Z_{q}^{PF}\left( \beta ,N\right)&=\sum _{\uplambda _{I,1}\le q}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.2)

where $l_{\left( \uplambda \right) }$ is the number of elements in the integer partition $\left( \uplambda \right)$ and $\uplambda _{I,1}$ is the largest element in $\left( \uplambda \right)$. 

In this section, we discuss parastatistics in the scheme. By expressing the canonical partition function in terms of the M-function and analysing the coefficient, we give the maximum occupation number of the para-particles. We show that the para-particle are not indistinguishable. The permutation phase for parastatistics is discussed in Refs. [4, 19, 27]. Our method is new and gives results such as the dimension of the permutation phase of the wave function.

5.1.1 The Maximum Occupation Number and the Indistinguishability of Para-Particles

It shows in Sects. 3.1 and 3.2 that if the canonical partition function is written as linear combinations of the M-function, the coefficient gives the indistinguishability and the maximum occupation number of particles. Here, with Eq. (4.1), the canonical partition function of parastatistics, Eqs. (5.1) and (5.2), can be written in terms of the M-function

$$\begin{aligned} Z_{q}^{PB}\left( \beta ,N\right)&=\sum _{I,J=1}^{P\left( N\right) } C_{b}^{\left( \uplambda \right) _{J}}\left( q\right) k_{J}^{I}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.3)

$$\begin{aligned} Z_{q}^{PF}\left( \beta ,N\right)&=\sum _{I,J=1}^{P\left( N\right) } C_{f}^{\left( \uplambda \right) _{J}}\left( q\right) k_{J}^{I}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.4)

where

$$\begin{aligned} C_{b}^{\left( \uplambda \right) _{J}}\left( q\right)&=\left\{ \begin{array}{l} 1\text {, for }l_{\left( \uplambda \right) _{J}}\le q,\\ 0\text {, otherwise} \end{array} \right. , \end{aligned}$$

(5.5)

$$\begin{aligned} C_{f}^{\left( \uplambda \right) _{J}}\left( q\right)&=\left\{ \begin{array}{l} 1\text {, for }\uplambda _{J,1}\le q,\\ 0\text {, otherwise} \end{array} \right. . \end{aligned}$$

(5.6)

In Eqs. (5.3) and (5.4), the coefficient of the M-function reads $\sum _{I=1}^{P\left( N\right) }C_{b}^{\left( \uplambda \right) _{J}}\left( q\right) k_{J}^{I}$ and $\sum _{I=1}^{P\left( N\right) }C_{f}^{\left( \uplambda \right) _{J}}\left( q\right) k_{J}^{I}$. For the sake of clarity, we list some explicit expressions of the canonical partition function, Eqs. (5.3) and (5.4), of $N=3$, 4, 5 with different q (for $q=1$ parastatistics recovers Bose and Fermi statistics). The detail of the calculation can be found in appendixes.

$N=3$, for para-Bose cases,

$$\begin{aligned} Z_{2}^{PB}\left( \beta ,3\right)= & {} m_{\left( 3\right) }+2m_{\left( 2,1\right) }+3m_{\left( 1^{3}\right) }, \end{aligned}$$

(5.7)

$$\begin{aligned} Z_{3}^{PB}\left( \beta ,3\right)= & {} m_{\left( 3\right) }+2m_{\left( 2,1\right) }+3m_{\left( 1^{3}\right) }. \end{aligned}$$

(5.8)

For para-Fermi cases,

$$\begin{aligned} Z_{2}^{PF}\left( \beta ,3\right)= & {} m_{\left( 2,1\right) }+3m_{\left( 1^{3}\right) }, \end{aligned}$$

(5.9)

$$\begin{aligned} Z_{3}^{PF}\left( \beta ,3\right)= & {} m_{\left( 3\right) }+2m_{\left( 2,1\right) }+4m_{\left( 1^{3}\right) }. \end{aligned}$$

(5.10)

$N=4$, for para-Bose cases,

$$\begin{aligned} Z_{2}^{PB}\left( \beta ,4\right)= & {} m_{\left( 4\right) }+2m_{\left( 3,1\right) }+3m_{\left( 2^{2}\right) }+4m_{\left( 2,1^{2}\right) }+6m_{\left( 1^{4}\right) }, \end{aligned}$$

(5.11)

$$\begin{aligned} Z_{3}^{PB}\left( \beta ,4\right)= & {} m_{\left( 4\right) }+2m_{\left( 3,1\right) }+3m_{\left( 2^{2}\right) }+5m_{\left( 2,1^{2}\right) }+9m_{\left( 1^{4}\right) }, \end{aligned}$$

(5.12)

$$\begin{aligned} Z_{4}^{PB}\left( \beta ,4\right)= & {} m_{\left( 4\right) }+2m_{\left( 3,1\right) }+3m_{\left( 2^{2}\right) }+5m_{\left( 2,1^{2}\right) }+10m_{\left( 1^{4}\right) }. \end{aligned}$$

(5.13)

For para-Fermi cases,

$$\begin{aligned} Z_{2}^{PF}\left( \beta ,4\right)= & {} m_{\left( 2^{2}\right) }+2m_{\left( 2,1^{2}\right) }+6m_{\left( 1^{4}\right) }, \end{aligned}$$

(5.14)

$$\begin{aligned} Z_{3}^{PF}\left( \beta ,4\right)= & {} m_{\left( 3,1\right) }+2m_{\left( 2^{2}\right) }+4m_{\left( 2,1^{2}\right) }+9m_{\left( 1^{4}\right) }, \end{aligned}$$

(5.15)

$$\begin{aligned} Z_{4}^{PF}\left( \beta ,4\right)= & {} m_{\left( 4\right) }+2m_{\left( 3,1\right) }+3m_{\left( 2^{2}\right) }+5m_{\left( 2,1^{2}\right) }+10m_{\left( 1^{4}\right) }. \end{aligned}$$

(5.16)

$N=5$, for para-Bose cases,

$$\begin{aligned} Z_{2}^{PB}\left( \beta ,5\right) =&\, m_{\left( 5\right) }+2m_{\left( 4,1\right) }+3m_{\left( 3,2\right) }+4m_{\left( 3,1^{2}\right) }\nonumber \\&+5m_{\left( 2^{2},1\right) }+7m_{\left( 2,1^{3}\right) }+10m_{\left( 1^{5}\right) }, \end{aligned}$$

(5.17)

$$\begin{aligned} Z_{3}^{PB}\left( \beta ,5\right) =&\, m_{\left( 5\right) }+2m_{\left( 4,1\right) }+3m_{\left( 3,2\right) }+5m_{\left( 3,1^{2}\right) }\nonumber \\&+7m_{\left( 2^{2},1\right) }+12m_{\left( 2,1^{3}\right) }+21m_{\left( 1^{5}\right) }, \end{aligned}$$

(5.18)

$$\begin{aligned} Z_{4}^{PB}\left( \beta ,5\right) =&\,m_{\left( 5\right) }+2m_{\left( 4,1\right) }+3m_{\left( 3,2\right) }+5m_{\left( 3,1^{2}\right) }\nonumber \\&+7m_{\left( 2^{2},1\right) }+13m_{\left( 2,1^{3}\right) }+25m_{\left( 1^{5}\right) }, \end{aligned}$$

(5.19)

$$\begin{aligned} Z_{5}^{PB}\left( \beta ,5\right) =\,&m_{\left( 5\right) }+2m_{\left( 4,1\right) }+3m_{\left( 3,2\right) }+5m_{\left( 3,1^{2}\right) }\nonumber \\&+7m_{\left( 2^{2},1\right) }+13m_{\left( 2,1^{3}\right) }+26m_{\left( 1^{5}\right) }. \end{aligned}$$

(5.20)

For para-Fermi cases,

$$\begin{aligned} Z_{2}^{PF}\left( \beta ,5\right)= & {} m_{\left( 2^{2},1\right) }+3m_{\left( 2,1^{3}\right) }+10m_{\left( 1^{5}\right) }, \end{aligned}$$

(5.21)

$$\begin{aligned} Z_{3}^{PF}\left( \beta ,5\right)= & {} m_{\left( 3,2\right) }+2m_{\left( 3,1^{2}\right) }+4m_{\left( 2^{2},1\right) }+9m_{\left( 2,1^{3}\right) }+21m_{\left( 1^{5}\right) }, \end{aligned}$$

(5.22)

$$\begin{aligned} Z_{4}^{PF}\left( \beta ,5\right) =&\, m_{\left( 4,1\right) }+2m_{\left( 3,2\right) }+4m_{\left( 3,1^{2}\right) }+6m_{\left( 2^{2},1\right) }\nonumber \\&+12m_{\left( 2,1^{3}\right) }+25m_{\left( 1^{5}\right) }, \end{aligned}$$

(5.23)

$$\begin{aligned} Z_{5}^{PF}\left( \beta ,5\right) =&\, m_{\left( 5\right) }+2m_{\left( 4,1\right) }+3m_{\left( 3,2\right) }+5m_{\left( 3,1^{2}\right) }\nonumber \\&+7m_{\left( 2^{2},1\right) }+13m_{\left( 2,1^{3}\right) }+26m_{\left( 1^{5}\right) }. \end{aligned}$$

(5.24)

The indistinguishability of para-particles. From Eqs. (5.7)–(5.24), one can see that there are coefficients of M-functions larger than 1, which means that exchanging two para-particles occupying different quantum states will lead to new microstates. Therefore, the para-particle is not indistinguishable.

The maximum occupation number of para-particles. As shown in Eqs. (5.7)–(5.24),since the Kostka number is a lower triangular matrix [22, 24], for para-Fermi statistics, the first non-zero coefficient in the canonical partition function gives $m_{\left( \uplambda \right) ^{q}}\left( e^{-\beta \varepsilon _{1}} ,e^{-\beta \varepsilon _{2}},\ldots \right)$ with $\uplambda _{I,1}=q$. For the para-Bose statistics, the first non-zero coefficient in the canonical partition function gives $m_{\left( N\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$ with $\uplambda _{I,1}=N$. That is, Eqs. (5.3) and (5.4) can be written in the form

$$\begin{aligned} Z_{q}^{PB}\left( \beta ,N\right)&=m_{\left( N\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) +\sum _{\uplambda _{I,1}<N}^{P\left( N\right) }M^{\prime I}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.25)

$$\begin{aligned} Z_{q}^{PF}\left( \beta ,N\right)&=m_{\left( \uplambda \right) ^{q}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) +\sum _{\uplambda _{I,1}<q}^{P\left( N\right) }M^{\prime \prime I}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(5.26)

Therefore, for para-Fermi statistics with the parameter q, the maximum occupation number is q. For para-Bose statistics, there is no limitation on the maximum occupation number.

Moreover, for para-Fermi statistics with the parameter q, although the maximum occupation number is q, it does not yield Gentile statistics. The indistinguishability of particles distinguishes those two kinds of statistics, e.g., $Z_{2}^{PF}\left( \beta ,5\right) =m_{\left( 2^{2},1\right) }+3m_{\left( 2,1^{3}\right) }+10m_{\left( 1^{5}\right) }$ and $Z_{2}^{G}\left( \beta ,5\right) =m_{\left( 2^{2},1\right) }+m_{\left( 2,1^{3}\right) }+m_{\left( 1^{5}\right) }$.

5.1.2 The Dimension of Permutation Phases of Para-Particles

The Hilbert subspace that describes the para-particles. The canonical partition function of parastatitics, Eqs. (5.1) and (5.2), shows that the Hilbert subspace describing para-Bose statistics is a direct sum of those Hilbert spaces corresponding to $\left( \uplambda \right)$ with length smaller than q. The Hilbert subspace describing para-Fermi statistics is a direct sum of those Hilbert spaces corresponding to $\left( \uplambda \right)$ with $\uplambda _{1}$ smaller than q, as shown in Figs. 3 and 4. That is,

$$\begin{aligned} D=\left\{ \begin{array}{c} {\displaystyle \bigoplus \limits _{l_{\left( \uplambda \right) }\le q}} V^{\left( \uplambda \right) }\text { for para-Bose statistics with the parameter }q,\\ {\displaystyle \bigoplus \limits _{\uplambda _{I,1}\le q}} V^{\left( \uplambda \right) }\text { for para-Fermi statistics with the parameter }q, \end{array} \right. \end{aligned}$$

(5.27)

as shown in Figs. 5 and 6.

Fig. 3

The Hilbert subspace describing para-Fermi statistics with q=2

Fig. 4

The Hilbert subspace describing para-Fermi statistics with q=3

Fig. 5

The Hilbert subspace describing para-Bose statistics with q=2

Fig. 6

The subspace describing para-Bose statistics with q=3

The permutation phase and the dimension of the permutation phase of the para-particles. For parastatistics, the permutation phase of the wave function can be given as

$$\begin{aligned} \sigma _{ij}|\Phi \rangle =D\left( \sigma _{ij}\right) |\Phi \rangle , \end{aligned}$$

(5.28)

where

$$\begin{aligned} D\left( \sigma _{ij}\right) =\left\{ \begin{array}{c} {\displaystyle \bigoplus \limits _{l_{\left( \uplambda \right) }\le q}} D^{I}\left( \sigma _{ij}\right) \text { for para-Bose statistics with parameter }q,\\ {\displaystyle \bigoplus \limits _{\uplambda _{I,1}\le q}} D^{I}\left( \sigma _{ij}\right) \text { for para-Fermi statistics with parameter }q. \end{array} \right. \end{aligned}$$

(5.29)

This is a multi-dimensional representation. By using the dimension of the representation of the permutation group [33], we give the dimension of the permutation phase. For para-Bose statistics with the parameter q,

$$\begin{aligned} \text {dim}\left[ D\left( \sigma _{ij}\right) \right] =\sum _{l_{\left( \uplambda \right) _{I}}\le q}N! {\displaystyle \prod \limits _{i=1,i<j}} \left( \uplambda _{I,i}-\uplambda _{I,j}-i+j\right) {\displaystyle \prod \limits _{i=1}} \left[ \left( l_{\left( \uplambda \right) }+\uplambda _{I,i}-i\right) !\right] ^{-1}.\qquad \end{aligned}$$

(5.30)

For para-Fermi statistics with the parameter q,

$$\begin{aligned} \text {dim}\left[ D\left( \sigma _{ij}\right) \right] =\sum _{\uplambda _{I,1}\le q}N! {\displaystyle \prod \limits _{i=1,i<j}} \left( \uplambda _{I,i}-\uplambda _{I,j}-i+j\right) {\displaystyle \prod \limits _{i=1}} \left[ \left( l_{\left( \uplambda \right) }+\uplambda _{I,i}-i\right) !\right] ^{-1}.\nonumber \\ \end{aligned}$$

(5.31)

For example, for a three-para-boson system with the parameter $q=2$, the Hilbert subspace is

$$\begin{aligned} \left. D\right| _{N=3,q=2}=V^{\left( 3\right) }\oplus V^{\left( 2,1\right) }, \end{aligned}$$

(5.32)

the permutation phase is

$$\begin{aligned} \left. D\left( \sigma _{ij}\right) \right| _{N=3,q=2}=\left( \begin{array}{cc} D^{\left( 3\right) }\left( \sigma _{ij}\right) &{} 0\\ 0 &{} D^{\left( 2,1\right) }\left( \sigma _{ij}\right) \end{array} \right) , \end{aligned}$$

(5.33)

and the dimension of the permutation phase is

$$\begin{aligned} \text {dim}\left[ \left. D\left( \sigma _{ij}\right) \right| _{N=3,q=2} \right] =\text {dim}\left[ D^{\left( 3\right) }\left( \sigma _{ij}\right) \right] +\text {dim}\left[ D^{\left( 2,1\right) }\left( \sigma _{ij}\right) \right] =1+2=3.\qquad \end{aligned}$$

(5.34)

For a four-para-boson system with the parameter $q=2$, the Hilbert subspace is

$$\begin{aligned} \left. D\right| _{N=4,q=2}=V^{\left( 4\right) }\oplus V^{\left( 3,1\right) }\oplus V^{\left( 2^{2}\right) }, \end{aligned}$$

(5.35)

the permutation phase is

$$\begin{aligned} \left. D\left( \sigma _{ij}\right) \right| _{N=4,q=2}=\left( \begin{array}{ccc} D^{\left( 4\right) }\left( \sigma _{ij}\right) &{} 0 &{} 0\\ 0 &{} D^{\left( 3,1\right) }\left( \sigma _{ij}\right) &{} 0\\ 0 &{} 0 &{} D^{\left( 2^{2}\right) }\left( \sigma _{ij}\right) \end{array} \right) , \end{aligned}$$

(5.36)

and the dimension of the permutation phase is

$$\begin{aligned} \text {dim}\left[ \left. D\left( \sigma _{ij}\right) \right| _{N=4,q=2} \right]&=\text {dim}\left[ D^{\left( 4\right) }\left( \sigma _{ij}\right) \right] +\text {dim}\left[ D^{\left( 3,1\right) }\left( \sigma _{ij}\right) \right] +\text {dim}\left[ D^{\left( 2^{2}\right) }\left( \sigma _{ij}\right) \right] \nonumber \\&=1+3+2=6. \end{aligned}$$

(5.37)

5.2 The N-Distinguishable-Particle Gas System: Boltzmann Statistics

The system consisting of distinguishable particles obeys Boltzmann statistics. The canonical partition function for an ideal N-distinguishable-particle gas is [1, 2]

$$\begin{aligned} Z_{cl}\left( \beta ,N\right) =\left( \sum _{i}e^{-\beta \varepsilon _{i} }\right) ^{N}. \end{aligned}$$

(5.38)

In this section, by discussing the N-distinguishable-particle gas in the scheme, we suggest an unconventional perspective to study the Hilbert subspace, the maximum occupation number, and the distinguishability of Boltzmann particles.

5.2.1 The Hilbert Space Describing Boltzmann Particles

One can verify that Eq. (5.38) can be expressed as a linear combination of the S-function:

$$\begin{aligned} Z_{cl}\left( \beta ,N\right) =\left( \sum _{i}e^{-\beta \varepsilon _{i} }\right) ^{N}=\sum _{I=1}^{P\left( N\right) }f_{I}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.39)

where the coefficient of the S-function $f_{I}$ is defined in Eq. (2.21). Eq. (5.39) implies that the Hilbert space describing the N -distinguishable-particle system is

$$\begin{aligned} D= {\displaystyle \bigoplus \limits _{I=1}^{P\left( N\right) }} \left( V^{\prime \left( \uplambda \right) _{I}}\right) ^{\oplus f_{I}}= {\displaystyle \bigoplus \limits _{I=1}^{P\left( N\right) }} V^{\left( \uplambda \right) _{I}}=V^{\otimes N}, \end{aligned}$$

(5.40)

where we use the fact that $V^{\prime \left( \uplambda \right) _{I}}$ occurs $f_{I}$ times in $V^{\left( \uplambda \right) _{I}}$, i.e., $V^{\left( \uplambda \right) _{I}}=\left( V^{\prime \left( \uplambda \right) _{I}}\right) ^{\oplus f_{I}}$. Therefore, as shown in Fig. 7, the Hilbert space $V^{\otimes N}$ describes an N-distinguishable-particle gas system.

Fig. 7

The Hilbert space describing the N-distinguishable-particle gas system. The $V^{\otimes N}$ describes the N-distinguishable-particle gas system

5.2.2 The Maximum Occupation Number and the Distinguishability of Boltzmann Particles

One can verify that Eq. (5.38) can be expressed as a linear combination of the M-function:

$$\begin{aligned} Z_{cl}\left( \beta ,N\right) =\left( \sum _{i}e^{-\beta \varepsilon _{i} }\right) ^{N}=\sum _{I=1}^{P\left( N\right) }z^{\left( \uplambda \right) _{I}}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1} },e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.41)

where the coefficient of the M-function $z^{\left( \uplambda \right) _{I}}$ satisfies

$$\begin{aligned} z^{\left( \uplambda \right) _{I}}=N!\left( {\displaystyle \prod \limits _{j=1}^{N}} \uplambda _{I,j}!\right) ^{-1}. \end{aligned}$$

(5.42)

The maximum occupation number of Boltzmann particles. One can find that there is no limitation on the maximum occupation number for an N-distinguishable-particle gas system, since the first term in Eqs. (5.41) is $m_{\left( N\right) }\left( e^{-\beta \varepsilon _{1} },e^{-\beta \varepsilon _{2}},\ldots \right)$.

The distinguishability of Boltzmann particles. A direct manifestation of the distinguishability of the particle is given: the coefficient $z^{\left( \uplambda \right) _{I}}$ appears because the number of microstates with $\uplambda _{I,1}$ distinguishable particles occupying a quantum state, $\uplambda _{I,2}$ distinguishable particles occupying another quantum state, and so on, is exactly $z^{\left( \uplambda \right) _{I}}$.

5.3 Gentile Statistics

Gentile statistics is a generalization of Bose–Einstein and Fermi–Dirac statistics. The maximum occupation number of Gentile statistics is an integer q [11, 12, 34]. The canonical partition function of an ideal Gentile gas is [26]

$$\begin{aligned} Z_{q}^{G}\left( \beta ,N\right) =\sum _{\uplambda _{I,1}\le q}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(5.43)

In this section, we give a discussion on Gentile statistics in the scheme. Especially, we show that the Hamiltonian of a Gentile-statistics system is noninvariant under permutations, therefore, the Gentile particle is not indistinguishable and the permutation phase of the wave function of Gentile particles can not be constructed in the scheme.

5.3.1 Discussions on the Permutation Phase of Gentile Particles

It shows in Sects. 3.3 and 3.4 that if the canonical partition function is written as linear combinations of the S-function, the coefficient gives the the permutation phase of wave function and the commutation relation between the Hamiltonian and the permutation group. Here, by using Eq (4.1), the canonical partition function of Gentile statistics, Eqs. (5.43), can be written in terms of the S-function:

$$\begin{aligned} Z_{q}^{G}\left( \beta ,N\right) =\sum _{I,J=1}^{P\left( N\right) }\Gamma ^{J}\left( q\right) \left( k_{J}^{I}\right) ^{-1}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2} },\ldots \right) , \end{aligned}$$

(5.44)

where $\left( k_{J}^{I}\right) ^{-1}$ satisfies $\sum _{I=1}^{P\left( N\right) }\left( k_{J}^{I}\right) ^{-1}k_{I}^{L}=\delta _{J}^{L}$ with $\delta _{J}^{L}$ the Kronecker delta function and $\Gamma ^{J}\left( q\right)$ satisfies

$$\begin{aligned} \Gamma ^{J}\left( q\right) =\left\{ \begin{array}{c} 1,\text {if }\uplambda _{I,1}\leqslant q\\ 0\text {, otherwise} \end{array} \right. . \end{aligned}$$

(5.45)

In Eq. (5.44) the coefficient of the S-function reads $\sum _{J=1}^{P\left( N\right) }\Gamma ^{J}\left( q\right) \left( k_{J}^{I}\right) ^{-1}$. For the sake of clarity, we list some explicit expressions of the canonical partition function, Eq. (5.44), of $N=3$, 4, 5, and 6 with different maximum occupation number q. Those results are in consistency with that in the previous work [26].

$N=3$,

$$\begin{aligned} Z_{2}^{G}\left( \beta ,3\right) =s_{\left( 2,1\right) }-s_{\left( 1^{3}\right) }, \end{aligned}$$

(5.46)

where we denote $s_{\left( \uplambda \right) }=s_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$ for convenience.

$N=4$,

$$\begin{aligned} Z_{2}^{G}\left( \beta ,4\right)= & {} s_{\left( 2^{2}\right) }-s_{\left( 1^{4}\right) }, \end{aligned}$$

(5.47)

$$\begin{aligned} Z_{3}^{G}\left( \beta ,4\right)= & {} s_{\left( 3,1\right) }-s_{\left( 2,1^{2}\right) }+s_{\left( 1^{4}\right) }. \end{aligned}$$

(5.48)

$N=5$,

$$\begin{aligned} Z_{2}^{G}\left( \beta ,5\right)= & {} s_{\left( 2^{2},1\right) }-s_{\left( 2,1^{3}\right) }, \end{aligned}$$

(5.49)

$$\begin{aligned} Z_{3}^{G}\left( \beta ,5\right)= & {} s_{\left( 3,2\right) }-s_{\left( 2^{2},1\right) }+s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.50)

$$\begin{aligned} Z_{4}^{G}\left( \beta ,5\right)= & {} s_{\left( 4,1\right) }-s_{\left( 3,1^{2}\right) }+s_{\left( 2,1^{3}\right) }-s_{\left( 1^{5}\right) }. \end{aligned}$$

(5.51)

$N=6$,

$$\begin{aligned} Z_{2}^{G}\left( \beta ,6\right)= & {} s_{\left( 2^{3}\right) }-s_{\left( 2,1^{4}\right) }+s_{\left( 1^{6}\right) }, \end{aligned}$$

(5.52)

$$\begin{aligned} Z_{3}^{G}\left( \beta ,6\right)= & {} s_{\left( 3,3\right) }-s_{\left( 2^{2},1^{2}\right) }+s_{\left( 2,1^{4}\right) }, \end{aligned}$$

(5.53)

$$\begin{aligned} Z_{4}^{G}\left( \beta ,6\right)= & {} s_{\left( 4,2\right) }-s_{\left( 3,2,1\right) }+s_{\left( 2^{2},1^{2}\right) }-s_{\left( 1^{6}\right) }, \end{aligned}$$

(5.54)

$$\begin{aligned} Z_{5}^{G}\left( \beta ,6\right)= & {} s_{\left( 5,1\right) }-s_{\left( 4,1^{2}\right) }+s_{\left( 3,1^{3}\right) }-s_{\left( 2,1^{4}\right) }+s_{\left( 1^{6}\right) }. \end{aligned}$$

(5.55)

The indistinguishability of Gentile particles. From Eqs. (5.46)–(5.55), one can see that there are negative coefficients of S-functions. Therefore, the Hamiltonian of Gentile particles is noninvariant under permutations, that is, the Gentile particles is not indistinguishable.

The permutation phase and the Hilbert subspaces of Gentile particles. Since the Hamiltonian of Gentile particles is noninvariant under permutation, the permutation phase of wave function for Gentile particles can not be constructed in the scheme. However, the Hilbert subspace describing Gentile statistics is the subspaces with nonzero coefficient, as shown in Figs. 8, 9 and 10.

Fig. 8

The Hilbert subspace describing Gentile statistics with maximum occupation number q = 2. The cube with horizontal lines means that the corresponding coefficient is negative

Fig. 9

The Hilbert subspace describing Gentile statistics with maximum occupation number q = 3. The cube with horizontal lines means that the corresponding coefficient is negative

Fig. 10

The Hilbert subspace describing Gentile statistics with maximum occupation number q = 4. The cube with horizontal lines means that the corresponding coefficient is negative

5.4 Immannons and Gentileonic Statistics

Gentileonic statistics is proposed by Cattani and Fernandes in 1984 [10]. Immannons are proposed by Tichy in 2017 [9]. They both are generalized statistics corresponding to the higher dimensional representation of permutation groups [9, 10].

In this section, we discuss immannons and Gentileonic statistics in the scheme. We give the canonical partition function for the immanonns and Gentileonic statistics. The result shows that in statistical mechanics, the immannons and Gentileonic statistics are essentially equivalent.

5.4.1 The Canonical Partition Function of Immanonns and Gentileonic Particles

In this section, we give the canonical partition function of immanonns and Gentileonic statistics.

The canonical partition function of immanonns and Gentileonic particles labeled by $\left( \uplambda \right)$ is

$$\begin{aligned} Z_{\left( \uplambda \right) }^{IG}\left( \beta ,N\right) =s_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2} },\ldots \right) . \end{aligned}$$

(5.56)

Equation (5.56) shows that in statistical mechanics, the immanonns and Gentileonic particles share the same canonical partition function and thus are essentially the same statistics.

Here is the detail of the calculation of Eq. (5.56). Gentileonic statistics is related to the higher dimensional representation of $S_{N}$ and the wave function is give as [10, 17]

$$\begin{aligned} \left| \Psi _{\left( \uplambda \right) }\right\rangle =\frac{1}{\sqrt{f_{I} }}\left( \begin{array}{c} \Phi _{1}^{\left( \uplambda \right) }\\ \Phi _{2}^{\left( \uplambda \right) }\\ ...\\ \Phi _{f_{I}}^{\left( \uplambda \right) } \end{array} \right) , \end{aligned}$$

(5.57)

where $\Phi _{i}^{\left( \uplambda \right) }$ is the operator associated with the Young shapes [30] corresponding to the integer partition $\left( \uplambda \right)$. Since the operator associated with the Young shape is one of the constructions for the basis of the subspace that carries the irreducible representation for $S_{N}$ [30], that is, the subspace spanned by the wave function $\left| \Psi _{\left( \uplambda \right) }\right\rangle$ is $V^{\left( \uplambda \right) }$. The canonical partition function is

$$\begin{aligned} Z_{\left( \uplambda \right) }^{IG}\left( \beta ,N\right)&=\sum \left\langle \Psi _{\left( \uplambda \right) }\right| e^{-\beta H_{N}}\left| \Psi _{\left( \uplambda \right) }\right\rangle \end{aligned}$$

(5.58)

$$\begin{aligned}&=\frac{1}{f_{I}}\sum _{i=1}^{f_{I}}\left\langle \Phi _{i}^{\left( \uplambda \right) }\right| e^{-\beta H_{N}}\left| \Phi _{i}^{\left( \uplambda \right) }\right\rangle \nonumber \\&=s_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}} ,e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.59)

where $\left\langle \Phi _{i}^{\left( \uplambda \right) }\right| e^{-\beta H_{N}}\left| \Phi _{i}^{\left( \uplambda \right) }\right\rangle =\left\langle \Phi _{j}^{\left( \uplambda \right) }\right| e^{-\beta H_{N} }\left| \Phi _{j}^{\left( \uplambda \right) }\right\rangle =s_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2} },\ldots \right)$ is used. $\Phi _{i}^{\left( \uplambda \right) }$ and $\Phi _{j}^{\left( \uplambda \right) }$ give the equivalent and irreducible representation for $H_{N}$.

Immannons [9] is a kind of generalized statistics, of which, the inner product of the wave function gives the immanant, labeled by an integer partition $\left( \uplambda \right)$ [9]. It recovers Bose–Einstein statistics for $\left( \uplambda \right) =\left( N\right)$, and Fermi-Dirac statistics for $\left( \uplambda \right) =\left( 1^{N}\right)$. The wave function of immannons is [9]

$$\begin{aligned} \left| \Phi _{\left( \uplambda \right) }\right\rangle \propto \hat{P}_{\uplambda }\left| \psi _{1},\psi _{2},\ldots ,\psi _{N}\right\rangle , \end{aligned}$$

(5.60)

where

$$\begin{aligned} {\hat{P}}_{\uplambda }=\frac{\chi _{\left( \uplambda \right) }\left( e\right) }{N!}\sum _{\sigma \in S_{N}}\chi _{\left( \uplambda \right) }\left( \sigma \right) {\hat{Q}}\left( \sigma \right) \end{aligned}$$

(5.61)

with $\chi _{\left( \uplambda \right) }\left( \sigma \right)$ the simple characteristic of $\sigma$ and ${\hat{Q}}\left( \sigma \right)$ an operator satisfying [9]

$$\begin{aligned} {\hat{Q}}\left( \sigma \right) \left| \psi _{1},\psi _{2},\ldots ,\psi _{N}\right\rangle =\left| \psi _{\sigma _{1}},\psi _{\sigma _{2}},\ldots ,\psi _{\sigma _{N}}\right\rangle . \end{aligned}$$

(5.62)

The wave function, Eq. (5.60), is a construction of the basis for the subspace $V^{\prime \left( \uplambda \right) }$. Thus the wave function, Eq. (5.60), directly yields

$$\begin{aligned} Z^{IG}_{\left( \uplambda \right) }\left( \beta ,N\right) =\left\langle \Phi _{\left( \uplambda \right) }\right| e^{-\beta H_{N}}\left| \Phi _{\left( \uplambda \right) }\right\rangle =s_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(5.63)

5.4.2 Discussions on the Permutation Phase of Immanonns and Gentileonic Particles

The Hilbert subspace of immanonns and Gentileonic particles. There are only a few differences between the Hilbert subspace describing Gentileonic statistics and the immanonns: the subspace describing the Gentileonic statistics is $V^{\left( \uplambda \right) }$ and the subspace describing the immanonns is $V^{\prime \left( \uplambda \right) }$, i.e.,

$$\begin{aligned} D=\left\{ \begin{array}{l} V^{\left( \uplambda \right) }\text {, for Gentileonic statistics,}\\ V^{\prime \left( \uplambda \right) }\text {, for the immanonns.} \end{array} \right. \end{aligned}$$

(5.64)

We have shown that the subspace $V^{\prime \left( \uplambda \right) }$ occurs $f_{\left( \uplambda \right) }$ times in $V^{\left( \uplambda \right) }$, thus these two subspaces are essentially the same.

The permutation phase of immanonns and Gentileonic particles. The immanonns does not possess a permutation symmetry, because $V^{\prime \left( \uplambda \right) }$, spanned by wave function Eq. (5.60), does not carry a representation for $S_{N}$. However, for Gentileonic statistics the permutation phase of the wave function is

$$\begin{aligned} \sigma _{ij}|\Phi \rangle =D^{I}\left( \sigma _{ij}\right) |\Phi \rangle . \end{aligned}$$

(5.65)

Fig. 11

The Hilbert subspace describing immannons and Gentileonic statistics labeled by the integer partition (4,$1^{2}$)

5.4.3 The Maximun Occupation Number and the Indistinguishability of Immanonns and Gentileonic Particles

The maximun occupation number of immanonns and Gentileonic particles. We express the canonical partition function of immannons and the Gentileonic statistics, Eq. (5.56), in terms of the M-function:

$$\begin{aligned} Z_{\left( \uplambda \right) _{J}}^{IG}\left( \beta ,N\right)&=\sum _{K,I=1}^{P\left( N\right) }\delta _{J,K}k_{K}^{I}m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) \nonumber \\&=\sum _{I=1}^{P\left( N\right) }k_{J}^{I}m_{\left( \uplambda \right) _{I} }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(5.66)

In Eq. (5.66) the coefficient reads $k_{J}^{I}$. Since the Kostka number is a lower triangular matrix, i.e., $k_{J}^{I}=0$ if $I<J$ [22, 24], the first term of in Eq. (5.66) is always

$$\begin{aligned} Z_{\left( \uplambda \right) _{J}}^{IG}\left( \beta ,N\right) =m_{\left( \uplambda \right) _{J}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) +\ldots , \end{aligned}$$

(5.67)

which implies that the maximum occupation number for the immanonns and Gentileonic statistics is $\uplambda _{J,1}$.

The indistinguishability of immannons and the Gentileonic particles. The coefficient of the M-function in Eq. (5.44) is the Kostka number $k_{J}^{I}$ with fixed J. For $J=1$, the immannons and the Gentileonic particles recover bosons, because the canonical partition function, Eq. (5.66), recovers the canonical partition function of bosons, $Z_{\left( \uplambda \right) _{1} }^{IG}\left( \beta ,N\right) =\sum _{I=1}^{P\left( N\right) }m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$ or $Z_{\left( \uplambda \right) _{1}} ^{IG}\left( \beta ,N\right) =s_{\left( N\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$. Except for $J=1$, there are coefficients of M-functions larger than 1 (neither 1 nor 0), therefore, immannons and the Gentileonic particles are not indistinguishable. For example, the explicit expression of the canonical partition function for immannons and Gentileonic statistics labeled by the integer partition $\left( 3,1^{2}\right)$ is

$$\begin{aligned} Z_{\left( 3,1^{2}\right) }^{IG}\left( \beta ,N\right) =m_{\left( 3,1^{2}\right) }+m_{\left( 2^{2},1\right) }+3m_{\left( 2,1^{3}\right) }+6m_{\left( 1^{5}\right) }. \end{aligned}$$

(5.68)

5.5 Generalized Statistics Dual to Gentile Statistics: GD-ons

In this section, we propose a new kind of generalized statistics dual to Gentile statistics. For convenience, we denote the particle GD-ons. By dual we mean that the relation between GD-ons and Gentile statistics is an analog to the relation between para-Bose and para-Fermi statistics.

5.5.1 The Canonical Partition Function: The Definition of GD-ons

The definition of GD-ons. The canonical partition function for GD-ons labeled by the integer partition $\left( \uplambda \right)$ is

$$\begin{aligned} Z_{q}^{*}\left( N,\beta \right) \equiv \sum _{I=1,l_{\left( \uplambda \right) _{I}}\le q}^{P\left( N\right) }m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(5.69)

GD-ons recovers bosons when $q=N$.

For Gentile statistics, the canonical partition function is given in Eq. (5.44). From Eqs. (5.69) and (5.44), one can see that the relation between GD-ons and Gentile statistics is an analog to the relation between para-Bose and para-Fermi statistics of which the canonical partition function is given in Eqs. (5.1) and (5.2).

5.5.2 Discussions on the Permutation Phase of GD-ons

By using Eq (4.1), we can rewrite the canonical partition function of GD-ons, Eq. (5.69), as linear combinations of the S-function:

$$\begin{aligned} Z_{q}^{*}\left( N,\beta \right) =\sum _{I,J=1}^{P\left( N\right) } \Gamma ^{\prime J}\left( q\right) \left( k_{J}^{I}\right) ^{-1}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \end{aligned}$$

(5.70)

where

$$\begin{aligned} \Gamma ^{\prime J}\left( q\right) =\left\{ \begin{array}{l} 1,\text {if }l_{\left( \uplambda \right) _{I}}\le q\\ 0\text {, otherwise} \end{array} \right. . \end{aligned}$$

(5.71)

In Eq. (5.70), the coefficient of the S-function reads $\sum _{J=1}^{P\left( N\right) }\Gamma ^{\prime J}\left( q\right) \left( k_{J}^{I}\right) ^{-1}$. For the sake of clarity, we list some of the explicit expression of the canonical partition function, Eq. (5.70), for $N=5$:

$$\begin{aligned} Z_{1}^{*}\left( 5,\beta \right) =s_{\left( 5\right) }-s_{\left( 4,1\right) }+s_{\left( 3,1^{2}\right) }-s_{\left( 2,1^{3}\right) }+s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.72)

$$\begin{aligned} Z_{2}^{*}\left( 5,\beta \right) =s_{\left( 5\right) }-s_{\left( 4,1\right) }+2s_{\left( 2,1^{3}\right) }-3s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.73)

$$\begin{aligned} Z_{3}^{*}\left( 5,\beta \right) =s_{\left( 5\right) }-s_{\left( 2,1^{3}\right) }+3s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.74)

$$\begin{aligned} Z_{4}^{*}\left( 5,\beta \right) =s_{\left( 5\right) }-s_{\left( 1^{5}\right) }. \end{aligned}$$

(5.75)

From Eqs. (5.72)-(5.75), one can find that the coefficient of the S-function is not non-negative. Thus, for GD-ons, the Hamiltonian is noninvariant under permutations. The GD-ons is not indistinguishable. The permutation phase of GD-ons can not be constructed in the scheme.

However, we can give the Hilbert subspace describing GD-ons: the Hilbert subspace describing GD-ons is the $\left( \uplambda \right)$ labeled subspace $V^{\left( \uplambda \right) }$ with nonzero coefficients of the S-function $s_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right)$.

5.5.3 The Maximum Occupation Number of GD-ons

Since the first term in the canonical partition function of GD-ons, Eqs. (5.69), is $m_{\left( N\right) }\left( e^{-\beta \varepsilon _{1} },e^{-\beta \varepsilon _{2}},\ldots \right)$, i.e.,

$$\begin{aligned} Z_{q}^{*}\left( N,\beta \right) =m_{\left( N\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) +\sum _{I=2,l_{\left( \uplambda \right) _{I}}\le q}^{P\left( N\right) }m_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) , \qquad \end{aligned}$$

(5.76)

the maximum occupation number, is N, that is, there is no limitation on the maximum occupation number for GD-ons.

5.6 Generalized Statistics Corresponding to the M-Function: M-ons

We have shown that the S-function is closely related to the permutation phase and the M-function is closely related to the maximum occupation number. The generalized statistics corresponding to one single S-function is given as immannons and Gentileonic statistics. In this section, we propose a kind of generalized statistics corresponding to on single M-function. For the sake of convenience, we denote the particle M-ons.

5.6.1 The Canonical Partition Function: The Definition of M-ons

The definition of M-ons. The canonical partition function of M-ons labeled by the integer partition $\left( \uplambda \right)$ is

$$\begin{aligned} Z_{\left( \uplambda \right) }^{M}\left( \beta ,N\right) =m_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2} },\ldots \right) . \end{aligned}$$

(5.77)

The M-ons recover fermions when $\left( \uplambda \right) =\left( 1^{N}\right)$.

5.6.2 Discussions on the Permutation Phase of M-ons

By using Eq. (4.1), we can express the canonical partition function, Eq. (5.77), in terms of the S-function:

$$\begin{aligned} Z_{\left( \uplambda \right) _{J}}^{M}\left( \beta ,N\right)&=\sum _{K=1}^{P\left( N\right) }\sum _{I=1}^{P\left( N\right) }\delta _{J,K}\left( k_{K}^{I}\right) ^{-1}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) \nonumber \\&=\sum _{I=1}^{P\left( N\right) }\left( k_{J}^{I}\right) ^{-1}s_{\left( \uplambda \right) _{I}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(5.78)

The coefficient of the S-function in Eq. (5.78) reads $\left( k_{J}^{I}\right) ^{-1}$. For the sake of clarity, we give the explicit expression of the canonical partition function of M-ons labeled by $\left( \uplambda \right) =\left( 5\right)$, $\left( 4,1\right)$, ..., are

$$\begin{aligned} Z_{\left( 5\right) }^{M}\left( \beta ,5\right)= & {} s_{\left( 5\right) }-s_{\left( 4,1\right) }+s_{\left( 3,1^{2}\right) }-s_{\left( 2,1^{3}\right) }+s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.79)

$$\begin{aligned} Z_{\left( 4,1\right) }^{M}\left( \beta ,5\right)= & {} s_{\left( 4,1\right) }-s_{\left( 3,2\right) }-s_{\left( 3,1^{2}\right) }+s_{\left( 2^{2},1\right) }+s_{\left( 2,1^{3}\right) }-2s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.80)

$$\begin{aligned} Z_{\left( 3,2\right) }^{M}\left( \beta ,5\right)= & {} s_{\left( 3,2\right) }-s_{\left( 3,1^{2}\right) }-s_{\left( 2^{2},1\right) }+2s_{\left( 2,1^{3}\right) }-2s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.81)

$$\begin{aligned} Z_{\left( 3,1^{2}\right) }^{M}\left( \beta ,5\right)= & {} s_{\left( 3,1^{2}\right) }-s_{\left( 2^{2},1\right) }-s_{\left( 2,1^{3}\right) }+3s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.82)

$$\begin{aligned} Z_{\left( 2^{2},1\right) }^{M}\left( \beta ,5\right)= & {} s_{\left( 2^{2},1\right) }-2s_{\left( 2,1^{3}\right) }+3s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.83)

$$\begin{aligned} Z_{\left( 2,1^{3}\right) }^{M}\left( \beta ,5\right)= & {} s_{\left( 2,1^{3}\right) }-4s_{\left( 1^{5}\right) }. \end{aligned}$$

(5.84)

From Eqs. (5.79)–(5.84), one can find that the coefficient of S-function is not non-negative, thus, for M-ons, the Hamiltonian is noninvariant under permutations. The M-ons is not indistinguishable. The permutation phase for M-ons can not be constructed in the scheme.

However, the Hilbert subspace describing the M-ons is the space with nonzero coefficients.

5.6.3 The Maximum Occupation Number of M-ons

We find that the maximum occupation number is $q=\uplambda _{1}$. In the case of M-ons, the only counted microstate is that there are $\uplambda _{1}$ particles occupying a quantum states, $\uplambda _{2}$ particles occupying another quantum state, and so on.

5.7 Generalized Statistics Corresponding to the Power Sum Symmetric Function: P-ons

In this section, we propose a kind of generalized statistics corresponding to the power sum symmetric function $p_{\left( \uplambda \right) }$. The power sum symmetric function, like the M-function and the S-function, is another important class of the symmetric function. For the sake of convenience, we denote the particle P-ons.

5.7.1 The Canonical Partition Function: The Definition of P-ons

The definition of P-ons. The canonical partition function of P-ons labeled by the integer partition $\left( \uplambda \right)$ is

$$\begin{aligned} Z_{\left( \uplambda \right) }^{P}\left( \beta ,N\right) =p_{\left( \uplambda \right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2} },\ldots \right) \end{aligned}$$

(5.85)

with

$$\begin{aligned} p_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots ,x_{n}\right) = {\displaystyle \prod \limits _{i=1}^{l_{\left( \uplambda \right) }}} m_{\left( \uplambda _{i}\right) }\left( x_{1},x_{2},\ldots ,x_{n}\right) . \end{aligned}$$

(5.86)

Interestingly, one can verify that the P-ons recovers the distinguishable particle when $\left( \uplambda \right) =\left( 1^{N}\right)$.

5.7.2 Discussions on the Permutation Phase of P-ons

By using the relation between the power sum symmetric function and the S-function [35],

$$\begin{aligned} p_{\left( \uplambda \right) _{K}}\left( x_{1},x_{2},\ldots \right) =\sum _{I=1}^{P\left( N\right) }\chi _{K}^{I}s_{\left( \uplambda \right) _{I} }\left( x_{1},x_{2},\ldots \right) , \end{aligned}$$

(5.87)

where $\chi _{K}^{I}$ is the simple characteristic of $S_{N}$, we can express the canonical partition function, Eq. (5.85), in terms of the S-function:

$$\begin{aligned} Z_{\left( \uplambda \right) _{K}}^{P}\left( \beta ,N\right)&=p_{\left( \uplambda \right) _{K}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) \nonumber \\&=\sum _{I=1}^{P\left( N\right) }\chi _{K}^{I}s_{\left( \uplambda \right) _{I}}\left( x_{1},x_{2},\ldots \right) . \end{aligned}$$

(5.88)

For the sake of clarity, we give the explicit expression of the canonical partition function of P-ons labeled by $\left( \uplambda \right) =\left( 5\right)$, $\left( 4,1\right)$, ...,:

$$\begin{aligned} Z_{\left( 5\right) }^{p}\left( 5,\beta \right)= & {} s_{\left( 5\right) }-s_{\left( 4,1\right) }+s_{\left( 3,1^{2}\right) }-s_{\left( 2,1^{3}\right) }+s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.89)

$$\begin{aligned} Z_{\left( 4,1\right) }^{p}\left( 5,\beta \right)= & {} s_{\left( 5\right) }-s_{\left( 3,2\right) }+s_{\left( 2^{2},1\right) }-s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.90)

$$\begin{aligned} Z_{\left( 3,2\right) }^{p}\left( 5,\beta \right)= & {} s_{\left( 5\right) }-s_{\left( 4,1\right) }+s_{\left( 3,2\right) }-s_{\left( 2^{2},1\right) }+s_{\left( 2,1^{3}\right) }-s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.91)

$$\begin{aligned} Z_{\left( 3,1^{2}\right) }^{p}\left( 5,\beta \right)= & {} s_{\left( 5\right) }+s_{\left( 4,1\right) }-s_{\left( 3,2\right) }-s_{\left( 2^{2},1\right) }+s_{\left( 2,1^{3}\right) }+s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.92)

$$\begin{aligned} Z_{\left( 2^{2},1\right) }^{p}\left( 5,\beta \right)= & {} s_{\left( 5\right) }+s_{\left( 3,2\right) }-2s_{\left( 3,1^{2}\right) }+s_{\left( 2^{2},1\right) }+s_{\left( 1^{5}\right) }, \end{aligned}$$

(5.93)

$$\begin{aligned} Z_{\left( 2,1^{3}\right) }^{p}\left( 5,\beta \right)= & {} s_{\left( 5\right) }+s_{\left( 4,1\right) }-s_{\left( 3,2\right) }-s_{\left( 2^{2},1\right) }+s_{\left( 2,1^{3}\right) }+s_{\left( 1^{5}\right) }. \end{aligned}$$

(5.94)

From Eqs. (5.89)–(5.94), one can find that the coefficient of S-function is not non-negative, thus, for P-ons, the Hamiltonian is noninvariant under permutations. The P-ons is not indistinguishable. The permutation phase for P-ons can not be constructed in the scheme.

However, the Hilbert subspace describing the P-ons is the space with nonzero coefficients.

5.7.3 The Maximum Occupation Number of P-ons

By using Eq. (4.1), we can express the canonical partition function, Eq. (5.88), in terms of the M-function:

$$\begin{aligned} Z_{\left( \uplambda \right) _{K}}^{P}\left( \beta ,N\right)&=p_{\left( \uplambda \right) _{K}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) \nonumber \\&=\sum _{j=1}^{P\left( N\right) }\left( \sum _{I=1}^{P\left( N\right) }\chi _{K}^{I}k_{I}^{J}\right) m_{\left( \uplambda \right) _{J}}\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2}},\ldots \right) . \end{aligned}$$

(5.95)

The maximum occupation number can be obtained from the non-zero coefficient in Eq. (5.95). Based on the property of the simple characteristics of the $S_{N}$ and the Kostka number $k_{I}^{J}$, Eq. (5.95) can be written as

$$\begin{aligned} Z_{\left( \uplambda \right) _{K}}^{P}\left( \beta ,N\right) =m_{\left( N\right) }\left( e^{-\beta \varepsilon _{1}},e^{-\beta \varepsilon _{2} },\ldots \right) +\ldots . \end{aligned}$$

(5.96)

One can find that, for P-ons, there is no limitation on the maximum occupation number.

For example, the explicit expression of the canonical partition function for P-ons labeled by $\left( \uplambda \right) =\left( 5\right)$, $\left( 4,1\right)$, ..., are

$$\begin{aligned} Z_{\left( 5\right) }^{p}\left( 5,\beta \right)= & {} m_{\left( 5\right) }, \end{aligned}$$

(5.97)

$$\begin{aligned} Z_{\left( 4,1\right) }^{p}\left( 5,\beta \right)= & {} m_{\left( 5\right) }+m_{\left( 4,1\right) }, \end{aligned}$$

(5.98)

$$\begin{aligned} Z_{\left( 3,2\right) }^{p}\left( 5,\beta \right)= & {} m_{\left( 5\right) }+m_{\left( 3,2\right) }, \end{aligned}$$

(5.99)

$$\begin{aligned} Z_{\left( 3,1^{2}\right) }^{p}\left( 5,\beta \right)= & {} m_{\left( 5\right) }+2m_{\left( 4,1\right) }+m_{\left( 3,2\right) }+2m_{\left( 2^{2},1\right) }, \end{aligned}$$

(5.100)

$$\begin{aligned} Z_{\left( 2^{2},1\right) }^{p}\left( 5,\beta \right)= & {} m_{\left( 5\right) }+m_{\left( 4,1\right) }+2m_{\left( 3,2\right) }+2s_{\left( 2^{2},1\right) }, \end{aligned}$$

(5.101)

$$\begin{aligned} Z_{\left( 2,1^{3}\right) }^{p}\left( 5,\beta \right)= & {} m_{\left( 5\right) }+2m_{\left( 4,1\right) }+m_{\left( 3,2\right) }+2m_{\left( 3,1^{2}\right) }, \end{aligned}$$

(5.102)

$$\begin{aligned} Z_{\left( 1^{5}\right) }^{p}\left( 5,\beta \right) =&\, m_{\left( 5\right) }+5m_{\left( 4,1\right) }+10m_{\left( 3,2\right) }+20m_{\left( 3,1^{2}\right) }\nonumber \\&+30m_{\left( 2^{2},1\right) }+60m_{\left( 2,1^{3}\right) } +120m_{\left( 1^{5}\right) }. \end{aligned}$$

(5.103)

6 Conclusions and Outlooks

Although, fundamental particles such as electrons and photons are either bosons or fermions, the property of excitations such as phonons are different. The generalized statistics are candidates to describe those excitations. In this paper, for particles obeying various kinds of generalized statistics, including parastatistics proposed in 1952 by Green [7, 8], the intermediate statistics or Gentile statistics proposed in 1940 by Gentile Jr [11, 12], Gentileonic statistics proposed by Cattani and Fernandes in 1984 [10], and immannons proposed by Tichy in 2017 [9], we suggest a unified framework to describe them.

We reveal the connection between the permutation phase of the wave function and the maximum occupation number.

Generalized statistics are commonly considered as quantum statistics. That is, particles obeying generalized statistics are commonly considered as indistinguishable particles. We show that under the action of permutations, only Bose and Fermi statistics are quantum statistics. Particles obeying generalized statistics are not completely indistinguishable and thus are not quantum particles.

Moreover, we provide a general formula of canonical partition functions of ideal N-particle gases who obey various kinds of generalized statistics. We suggest a method to obtain the maximum occupation number and the permutation phase of the wave function, to identify the indistinguishability of particles, to determine the commutation relation between the Hamiltonian and the permutation group from the canonical partition function. We propose three new kinds of generalized statistics which seem to be the missing pieces in the puzzle.

The mathematical basis of the scheme is the mathematical theory of the invariant matrix [22], the Schur-Weyl duality [23], the symmetric function [22, 24], and the representation theory of the permutation group and the unitary group [30, 33]. Results in this paper together with our previous works [26, 36] build a bridge between the quantum statistical mechanics and such mathematical theories and enables one to use the fruitful result in the theory of the symmetric function to solve the problem in quantum statistical mechanics.

Various kinds of statistics considered here are generalized from a permutation-group point of view. Exchanging two particles of the system can also be described by other groups, such as the braid group. The braid group have a closed relation with the fundamentional group of the state space which relates to the dimensionality and topology of the system. We will work on this topic in later works.

Various kinds of statistics considered here are extensive statistics [37, 38], that is the probability of observing micro-states with energy E is proportional to $e^{-\beta E}$. In the nonextensive statistics [39], which is a generalization of Boltzmann-Gibbs statistical, the probability distribution is different from that in the extensive statistics, thus yielding variety of deformed entropies. For example, the Tsallis distribution and the Tsallis entropy [39]. It recently draws researchers attention to study the nonextensive statistics [40,41,42,43,44,45]. We will work on topics related to the non-extensivity and generalized statistics in later works.

7 Appendix

7.1 The Integer Partition and the Symmetric Function: A Brief Review

The main result of the present paper involves some basic knowledges of the mathematical theory of the integer partition and the symmetric function. For example, by arranging integer partitions in a prescribed order, we can better explain the main result. In this section, we give a brief review on theories of the partition function and the symmetric function. For more details, one can refer to Refs. [22, 24, 46] .

7.1.1 The Integer Partition

The integer partition and the length of an integer partition.An integer N can be represented as a sum of other integers:

$$\begin{aligned} N=\uplambda _{1}+\uplambda _{2}+\ldots +\uplambda _{l}, \end{aligned}$$

(7.1)

where $\uplambda _{1}\ge \uplambda _{2}\ge \ldots \ge \uplambda _{l}>0$. The integer partition of N is denoted by the notation $\left( \uplambda \right) =\left( \uplambda _{1},\uplambda _{2},\ldots ,\uplambda _{l}\right)$. The number of the integer in $\left( \uplambda \right)$ is the length of $\left( \uplambda \right)$, denoted by $l_{\left( \uplambda \right) }$. N is the size of $\left( \uplambda \right)$. For example, for an integer partition $\left( \uplambda \right) =\left( 3,2,1\right)$, the length is $l_{\left( \uplambda \right) }=3$ and the size is 6.

The unrestricted partition function P(N) and arranging integer partitions in a prescribed order. An integer N has many integer partitions and the unrestricted partition function P(N) counts the number of integer partitions [3]. For a given N, one arranges integer partitions in the following order: $\left( \uplambda \right)$, $\left( \uplambda \right) ^{\prime }$, when $\uplambda _{1}$ > $\uplambda _{1}^{\prime }$. $\left( \uplambda \right)$, $\left( \uplambda \right) ^{\prime }$, when $\uplambda _{1}=\uplambda _{1}^{\prime }$ but $\uplambda _{2}$ > $\uplambda _{2}^{\prime }$. and so on. One keeps comparing $\uplambda _{i}$ and $\uplambda _{i}^{\prime }$ until all integer partitions of N are arranged in a prescribed order. $\left( \uplambda \right) _{I}$ is the Ith integer partition function. For example, Integer partitions of 4 are $\left( \uplambda \right) _{1}=\left( 4\right)$, $\left( \uplambda \right) _{2}=\left( 3,1\right)$, $\left( \uplambda \right) _{3}=\left( 2^{2}\right)$, $\left( \uplambda \right) _{4}=\left( 2,1^{2}\right)$, and $\left( \uplambda \right) _{5}=\left( 1^{4}\right)$, where, e.g., the superscript in $1^{4}$ means 1 appearing four times, the superscript in $2^{2}$ means 2 appearing twice, and so on.

The conjugate integer partition. For an integer partition $\left( \uplambda \right) =\left( \uplambda _{1},\uplambda _{2},\ldots ,\uplambda _{l}\right)$, there is one and only one integer partition $\left( \uplambda \right) ^{*}$ that is conjugate to $\left( \uplambda \right)$. To get the conjugate integer partition $\left( \uplambda \right) ^{*}$ from $\left( \uplambda \right)$, an efficient method is to use the Young diagram [22, 24, 47]. For example, the conjugate integer partition of $\left( \uplambda \right) =\left( 3,1\right)$, as shown in Fig. 12, is $\left( \uplambda \right) ^{*}=\left( 2,1^{2}\right)$.

Fig. 12

An example of the method to obtain the conjugate integer partition by Young diagram

7.1.2 The Symmetric Function

The symmetric function, first studied by Hall in 1950s, is an important issue in algebraic combinatorics [22, 24]. It is closely related to integer partitions in number theory and plays an important role in the theory of group representations [22, 24, 46].

In this section, we give a brief review on several important types of symmetric functions, such as the S-function $s_{\left( \uplambda \right) }\left( x_{1},x_{2},x_{3},\ldots \right)$, the M-function $m_{\left( \uplambda \right) }\left( x_{1},x_{2},x_{3},\ldots \right)$, and the power sum symmetric function $p_{\left( \uplambda \right) }\left( x_{1} ,x_{2},x_{3},\ldots \right)$.

The S-function $s_{\left( \uplambda \right) }\left( x_{1},x_{2} ,x_{3},\ldots \right)$. The S-function, also called the Schur function, is an important type of the symmetric function. For an integer partition $\left( \uplambda \right)$ of the integer N, the S-function is defined by [22, 24]

$$\begin{aligned} s_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots ,x_{n}\right) =\frac{\det \left( x_{i}^{\uplambda _{j}+n-j}\right) }{\det \left( x_{i}^{n-j}\right) }, \end{aligned}$$

(7.2)

where $\det (A)$ represents the determinate of matrix A, $\uplambda _{j}$ is the jth element in the integer partition $\left( \uplambda \right)$. There is also another definition of $s_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots \right)$ without limitation on the number of variables [22, 24]:

$$\begin{aligned} s_{\left( \uplambda \right) _{I}}\left( x_{1},x_{2},\ldots \right) =\sum _{J=1}^{P\left( N\right) }\frac{g_{J}}{N!}\chi _{J}^{I} {\displaystyle \prod \limits _{m=1}^{k}} \left( \sum _{i}x_{i}^{m}\right) ^{a_{J,m}}, \end{aligned}$$

(7.3)

where $\left( \uplambda \right) _{I}$ is the Ith integer partition of the integer N and $a_{J,m}$ represents the times m occuring in $\left( \uplambda \right) _{J}$, $\chi _{J}^{I}$ is the simple characteristic of the permutation group of order N. $g_{I}$ satisfies

$$\begin{aligned} g_{I}=N!\left( {\displaystyle \prod \limits _{j=1}^{N}} j^{a_{I,j}}a_{I,j}!\right) ^{-1}. \end{aligned}$$

(7.4)

The monomial symmetric function: M-function $m_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots \right)$. For an integer partition $\left( \uplambda \right)$, the corresponding M-function is defined by [22, 24]

$$\begin{aligned} m_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots ,x_{n}\right) =\sum _{perm}x_{1}^{\uplambda _{1}}x_{2}^{\uplambda _{2}}\ldots ., \end{aligned}$$

(7.5)

where $\sum _{perm}$ indicates that the summation runs over all possible monotonically increasing permutations of $x_{i}$.

The power sum symmetric function $p_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots \right)$. For an integer partition $\left( \uplambda \right)$, the corresponding symmetric polynomial is defined by [22, 24]:

$$\begin{aligned} p_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots ,x_{n}\right) = {\displaystyle \prod \limits _{i=1}^{l_{\left( \uplambda \right) }}} m_{\left( \uplambda _{i}\right) }. \end{aligned}$$

(7.6)

The relation between $s_{\left( \uplambda \right) }\left( x_{1} ,x_{2},\ldots \right)$, $m_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots ,x_{n}\right)$, and $p_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots ,x_{n}\right)$. The S-function can be represented as a linear combination of the monomial symmetric polynomial [22, 24], i.e.,

$$\begin{aligned} s_{\left( \uplambda \right) _{K}}\left( x_{1},x_{2},\ldots \right) =\sum _{I=1}^{P\left( N\right) }k_{K}^{I}m_{\left( \uplambda \right) _{I}}\left( x_{1},x_{2},\ldots \right) , \end{aligned}$$

(7.7)

where $k_{K}^{I}$ is the Kostka number [22, 24]. The power sum symmetric function $p_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots \right)$ can be represented as a linear combination of the S-function $s_{\left( \uplambda \right) }\left( x_{1},x_{2},\ldots \right)$ [35]:

$$\begin{aligned} p_{\left( \uplambda \right) _{K}}\left( x_{1},x_{2},\ldots \right) =\sum _{I=1}^{P\left( N\right) }\chi _{K}^{I}s_{\left( \uplambda \right) _{I} }\left( x_{1},x_{2},\ldots \right) , \end{aligned}$$

(7.8)

where $\chi _{K}^{I}$ is the simple characteristics of $S_{N}$ [35, 46].

7.2 The Kostka Number

In this section, we give some Kostka numbers as examples.

The Kostka number $k_{K}^{J}$ is labeled by the Jth and the Kth integer partitions of N. Here we take $N=3,4,5$ as examples. For $N=3$, the Kostka number $k_{K}^{J}$ is $k_{\left( 3\right) }^{\left( 3\right) }=k_{1}^{1}=1$, $k_{\left( 2,1\right) }^{\left( 3\right) }=k_{2}^{1}=0$, $k_{\left( 1^{3}\right) }^{\left( 3\right) }=k_{3}^{1}=0$, $k_{\left( 3\right) }^{\left( 2,1\right) } =k_{1}^{2}=1$, $k_{\left( 2,1\right) }^{\left( 2,1\right) }=k_{2}^{2}=1$, $k_{\left( 1^{3}\right) }^{\left( 2,1\right) }=k_{3}^{2}=0$, $k_{\left( 3\right) }^{\left( 1^{3}\right) }=k_{1}^{3}=1$, $k_{\left( 2,1\right) }^{\left( 1^{3}\right) }=k_{2}^{3}=2$, and $k_{\left( 1^{3}\right) }^{\left( 1^{3}\right) }=k_{3}^{3}=1$ [22, 24]. For clarity , we rewrite the Kostka number in a matrix form:

$$\begin{aligned} k=\left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) , \end{aligned}$$

(7.9)

where we take the upper index of $k_{K}^{J}$ as the row index and the lower index as the column index.

For $N=4$, the Kostka number is [22, 24]

$$\begin{aligned} k=\left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) . \end{aligned}$$

(7.10)

For $N=5$, the Kostka number is [22, 24]

$$\begin{aligned} k=\left( \begin{array}{ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) . \end{aligned}$$

(7.11)

7.3 The Simple Characteristic of $S_{N}$

Here we take $N=3,4,5$ as examples. For $N=3$, The simple characteristic $\chi _{K}^{J}$ is $\chi _{\left( 3\right) }^{\left( 3\right) }=\chi _{1}^{1}=1$, $\chi _{\left( 2,1\right) }^{\left( 3\right) }=\chi _{2}^{1}=1$, $\chi _{\left( 1^{3}\right) }^{\left( 3\right) }=\chi _{3}^{1}=1$, $\chi _{\left( 3\right) }^{\left( 2,1\right) }=\chi _{1}^{2}=2$, $\chi _{\left( 2,1\right) }^{\left( 2,1\right) }=\chi _{2}^{2}=0$, $\chi _{\left( 1^{3}\right) }^{\left( 2,1\right) }=\chi _{3}^{2}=-1$, $\chi _{\left( 3\right) }^{\left( 1^{3}\right) }=\chi _{1}^{3}=1$, $\chi _{\left( 2,1\right) }^{\left( 1^{3}\right) }=\chi _{2}^{3}=-1$, and $\chi _{\left( 1^{3}\right) }^{\left( 1^{3}\right) }=\chi _{3}^{3}=1$ [30]. For clarity , we can rewrite the simple characteristic in a matrix form:

$$\begin{aligned} \chi =\left( \begin{array}{ccc} 1 &{}\quad 1 &{}\quad 1\\ 2 &{}\quad 0 &{}\quad -1\\ 1 &{}\quad -1 &{}\quad 1 \end{array} \right) . \end{aligned}$$

(7.12)

For $N=4$, $\chi$ is [30]

$$\begin{aligned} \chi =\left( \begin{array}{ccccc} 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ 3 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad -1\\ 2 &{}\quad -1 &{}\quad 2 &{}\quad 0 &{}\quad 0\\ 3 &{}\quad 0 &{}\quad -1 &{}\quad -1 &{}\quad 1\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad -1 &{}\quad -1 \end{array} \right) . \end{aligned}$$

(7.13)

For $N=5$, $\chi$ is [30]

$$\begin{aligned} \chi =\left( \begin{array}{ccccccc} 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ -1 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 4\\ 0 &{}\quad -1 &{}\quad 1 &{}\quad -1 &{}\quad 1 &{}\quad -1 &{}\quad 5\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -2 &{}\quad 0 &{}\quad 6\\ 0 &{}\quad 1 &{}\quad -1 &{}\quad -1 &{}\quad 1 &{}\quad -1 &{}\quad 5\\ -1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 4\\ 1 &{}\quad -1 &{}\quad -1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \end{array} \right) . \end{aligned}$$

(7.14)

7.4 The Calculation of the Maximum Occupation Number of the Parastatistics: An Example of Calculation the Coefficient in the canonical Partition Function

The canonical partition function of an N-particle gas under various kinds of generalized statistics can be expressed in terms of symmetric functions such as the S-function, the M-function, and so on. The procedure of the calculation of the coefficient in the canonical partition function involves the transformation of the coefficient by the matrix of the Kostka number or the simple characteristic. In this appendix, we give details of the calculation of the coefficient for parastatistics as an example.

When $N=3$, $q=1$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 0\\ 0 \end{array} \right) =\left( \begin{array}{c} 1\\ 1\\ 1 \end{array} \right) , \end{aligned}$$

(7.15)

that is,

$$\begin{aligned} Z_{1}^{PB}\left( \beta ,3\right)&=m_{\left( 3\right) } +m_{\left( 2,1\right) } +m_{\left( 1^{3}\right) }. \end{aligned}$$

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 0\\ 0\\ 1 \end{array} \right) =\left( \begin{array}{c} 0\\ 0\\ 1 \end{array} \right) , \end{aligned}$$

(7.16)

that is,

$$\begin{aligned} Z_{1}^{PB}\left( \beta ,3\right) =m_{\left( 1^{3}\right) }. \end{aligned}$$

The corresponding expression of the canonical partition function is already given in Eqs. (5.7)–(5.24), thus, in the following, we only give the coefficient.

When $N=3$ and $q=2$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 1\\ 0 \end{array} \right) =\left( \begin{array}{c} 1\\ 2\\ 3 \end{array} \right) . \end{aligned}$$

(7.17)

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 0\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{c} 0\\ 1\\ 3 \end{array} \right) . \end{aligned}$$

(7.18)

When $N=3$ and $q=3$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{c} 1\\ 2\\ 3 \end{array} \right) . \end{aligned}$$

(7.19)

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{c} 1\\ 2\\ 3 \end{array} \right) . \end{aligned}$$

(7.20)

When $N=4$ and $q=1$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 0\\ 0\\ 0\\ 0 \end{array} \right) =\left( \begin{array}{c} 1\\ 1\\ 1\\ 1\\ 1 \end{array} \right) . \end{aligned}$$

(7.21)

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 0\\ 0\\ 0\\ 0\\ 1 \end{array} \right) =\left( \begin{array}{c} 0\\ 0\\ 0\\ 0\\ 1 \end{array} \right) . \end{aligned}$$

(7.22)

When $N=4$ and $q=2$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 1\\ 1\\ 0\\ 0 \end{array} \right) =\left( \begin{array}{c} 1\\ 2\\ 3\\ 4\\ 6 \end{array} \right) . \end{aligned}$$

(7.23)

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 0\\ 0\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{c} 0\\ 0\\ 1\\ 2\\ 6 \end{array} \right) . \end{aligned}$$

(7.24)

When $N=4$ and $q=3$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 1\\ 1\\ 1\\ 0 \end{array} \right) =\left( \begin{array}{c} 1\\ 2\\ 3\\ 4\\ 9 \end{array} \right) . \end{aligned}$$

(7.25)

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 0\\ 1\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{c} 0\\ 1\\ 2\\ 4\\ 9 \end{array} \right) . \end{aligned}$$

(7.26)

When $N=4$ and $q=4$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 1\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{c} 1\\ 2\\ 3\\ 5\\ 10 \end{array} \right) . \end{aligned}$$

(7.27)

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array} \right) \left( \begin{array}{c} 1\\ 1\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{c} 1\\ 2\\ 3\\ 5\\ 10 \end{array} \right) . \end{aligned}$$

(7.28)

When $N=5$ and $q=1$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( \begin{array} {c} 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0 \end{array} \right) =\left( \begin{array} {c} 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1 \end{array} \right) . \end{aligned}$$

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( \begin{array} {c} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1 \end{array} \right) =\left( \begin{array} {c} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1 \end{array} \right) . \end{aligned}$$

When $N=5$ and $q=2$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0\\ 1 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0\\ 1 \quad 1 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0\\ 1 \quad 2 \quad 1 \quad 1 \quad 0 \quad 0 \quad 0\\ 1 \quad 2 \quad 2 \quad 1 \quad 1 \quad 0 \quad 0\\ 1 \quad 3 \quad 3 \quad 3 \quad 2 \quad 1 \quad 0\\ 1 \quad 4 \quad 5 \quad 6 \quad 5 \quad 4 \quad 1 \end{array} \right) \left( \begin{array} {c} 1\\ 1\\ 1\\ 0\\ 0\\ 0\\ 0 \end{array} \right) =\left( \begin{array} {c} 1\\ 2\\ 3\\ 4\\ 5\\ 7\\ 10 \end{array} \right) . \end{aligned}$$

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( F \begin{array} {c} 0\\ 0\\ 0\\ 0\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array} {c} 0\\ 0\\ 0\\ 0\\ 1\\ 3\\ 10 \end{array} \right) . \end{aligned}$$

When $N=5$ and $q=3$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( \begin{array} {c} 1\\ 1\\ 1\\ 1\\ 1\\ 0\\ 0 \end{array} \right) =\left( \begin{array} {c} 1\\ 2\\ 3\\ 5\\ 7\\ 12\\ 21 \end{array} \right) . \end{aligned}$$

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( \begin{array} {c} 0\\ 0\\ 1\\ 1\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array} {c} 0\\ 0\\ 1\\ 2\\ 4\\ 9\\ 21 \end{array} \right) . \end{aligned}$$

When $N=5$ and $q=4$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( \begin{array} {c} 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 0 \end{array} \right) =\left( \begin{array} {c} 1\\ 2\\ 3\\ 5\\ 7\\ 13\\ 25 \end{array} \right) . \end{aligned}$$

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( \begin{array} {c} 0\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array} {c} 0\\ 1\\ 2\\ 4\\ 6\\ 12\\ 25 \end{array} \right) . \end{aligned}$$

When $N=5$ and $q=5$, for para-Bose statistics, the coefficient in Eq. (5.3) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( \begin{array} {c} 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array} {c} 1\\ 2\\ 3\\ 5\\ 7\\ 13\\ 26 \end{array} \right) . \end{aligned}$$

For para-Fermi statistics, the coefficient in Eq. (5.4) is

$$\begin{aligned} \left( \begin{array} {ccccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 3 &{}\quad 3 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 5 &{}\quad 4 &{}\quad 1 \end{array} \right) \left( \begin{array} {c} 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1 \end{array} \right) =\left( \begin{array}{c} 1\\ 2\\ 3\\ 5\\ 7\\ 13\\ 26 \end{array} \right) . \end{aligned}$$