Parity distribution and divisibility of Mex-related partition functions

Abstract

Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(\lambda)$ for an integer partition $\lambda$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $\lambda$. They then defined $p_{A,a}(n)$ to be the number of partitions $\lambda$ of $n$ satisfying $\text{mex}_{A,a}(\lambda)\equiv a\pmod{2A}$. They found the generating function for $p_{t,t}(n)$ and $p_{2t,t}(n)$ for any positive integer $t$, and studied their arithmetic properties for some small values of $t$. In this article, we study the partition function $p_{mt,t}(n)$ for all positive integers $m$ and $t$. We show that for sufficiently large $X$, the number of all positive integer $n\leq X$ such that $p_{mt,t}(n)$ is an even number is at least $\mathcal{O}(\sqrt{X/3})$ for all positive integers $m$ and $t$. We also prove that for sufficiently large $X$, the number of all positive integer $n\leq X$ such that $p_{mp,p}(n)$ is an odd number is at least $\mathcal{O}(\log \log X)$ for all $m\not \equiv 0\pmod{3}$ and all primes $p\equiv 1\pmod{3}$. Finally, we establish identities connecting the ordinary partition function to $p_{mt,t}(n)$.