An easy upper bound for Ramsey numbers

TL;DR

This paper explores conjugacy growth series of symmetric and alternating groups, revealing connections to partition functions and proposing new graph classes, with implications for understanding group growth and number theory.

Contribution

It introduces a novel approach to compute conjugacy growth series for various groups and generating sets, linking them to partition functions and proposing the class of partition-complete graphs.

Findings

Conjugacy growth series of symmetric groups relate to partition functions.

Introduction of partition-complete graphs generalizing semi-hamiltonian graphs.

Numerical evidence of Ramanujan-like congruences in related series.

Abstract

It is observed that the conjugacy growth series of the infinite finitary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. Numerical evidences indicate that the coefficients of a series related to the finitary alternating group seem to satisfy congruence relations reminiscent of Ramanujan's congruences for the partition function.