A statistical mechanical approach to restricted integer partition functions

TL;DR

This paper introduces a novel statistical mechanical method to compute generating functions for restricted integer partition functions, linking them to quantum gases and symmetric functions, and generalizing Gentile statistics.

Contribution

It proposes a new approach using statistical mechanics to analyze restricted integer partitions and introduces a generalized partition function extending Gentile statistics.

Findings

Generated explicit expressions for restricted integer partition functions.

Linked generating functions to symmetric functions invariant under permutation groups.

Unified various types of restricted partitions within a generalized framework.

Abstract

The main aim of this paper is twofold: (1) Suggesting a statistical mechanical approach to the calculation of the generating function of restricted integer partition functions which count the number of partitions --- a way of writing an integer as a sum of other integers under certain restrictions. In this approach, the generating function of restricted integer partition functions is constructed from the canonical partition functions of various quantum gases. (2) Introducing a new type of restricted integer partition functions corresponding to general statistics which is a generalization of Gentile statistics in statistical mechanics; many kinds of restricted integer partition functions are special cases of this restricted integer partition function. Moreover, with statistical mechanics as a bridge, we reveals a mathematical fact: the generating function of restricted integer partition function is just the symmetric function which is a class of functions being invariant under the action of permutation groups. Using the approach, we provide some expressions of restricted integer partition functions as examples.