Asymptotics and congruences for partition functions which arise from finitary permutation groups

TL;DR

This paper derives asymptotic formulas and proves congruences for generalized partition functions related to finitary permutation groups, extending classical partition theory results and confirming conjectured congruences.

Contribution

It provides the first asymptotic formulas and proves over 200 conjectured congruences for generalized partition functions arising from finitary permutation groups.

Findings

Derived asymptotic formulas for all generalized partition functions p(n)_e.

Proved over 200 conjectured congruences for these functions.

Extended classical partition congruence results to new generalized functions.

Abstract

In a recent paper, Bacher and de la Harpe study the conjugacy growth series of finitary permutation groups. In the course of studying the coefficients of a series related to the finitary alternating group, they introduce generalized partition functions $p(n)_{\textbf{e}}$. The group theory motivates the study of the asymptotics for these functions. Moreover, Bacher and de la Harpe also conjecture over 200 congruences for these functions which are analogous to the Ramanujan congruences for the unrestricted partition function $p(n)$. We obtain asymptotic formulas for all of the $p(n)_{\textbf{e}}$ and prove their conjectured congruences.