On Mex-related partition functions of Andrews and Newman

TL;DR

This paper investigates the parity and congruence properties of specific partition functions related to the mex function, extending previous work and proving that these functions are mostly even with infinite families satisfying certain mod 2 congruences.

Contribution

It extends the analysis of mex-related partition functions to new parameter families and establishes their parity behavior and infinite congruence families using advanced number theory techniques.

Findings

Functions are almost always even for all considered parameters.

Established infinite families of congruences modulo 2 for these functions.

Extended previous parity results to broader classes of mex-related partition functions.

Abstract

The minimal excludant, or "mex" function, on a set $S$ of positive integers is the least positive integer not in $S$. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely $p_{t,t}(n)$, and provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$. In this article, we study the parity of $p_{t,t}(n)$ when $t=2^{\alpha}, 3\cdot 2^{\alpha}$ for all $\alpha\geq 1$. We prove that $p_{2^{\alpha},2^{\alpha}}(n)$ and $p_{3\cdot2^{\alpha}, 3\cdot2^{\alpha}}(n)$ are almost always even for all $\alpha\geq 1$. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we also find infinite families of congruences modulo $2$ satisfied by $p_{2^{\alpha},2^{\alpha}}(n)$ and $p_{3\cdot2^{\alpha}, 3\cdot2^{\alpha}}(n)$ for all $\alpha\geq 1$.