The Minimal excludant of $\ell$-regular partition

Abstract

Andrews and Newman \cite{andrewsnewmanI} introduced and studied ``mex,'' function for an integer partition $λ$ of a positive integer $n$. The related function $σ^{\mathrm{mex}}(n)$ was subsequently defined, and its generating function and other properties have been studied by several authors. In this article, for a positive integer $\ell \geq 2$, we define the minimal excludant of an $\ell$-regular partition $λ$ of a positive integer $n$. We then define $σ_\ell^{\mathrm{mex}}(n)$, the sum of the minimal excludants over all $\ell$-regular partitions of $n$, and obtain explicit generating functions for $σ_\ell^{\mathrm{mex}}(n)$ in the cases $\ell = 2$ and $\ell = 3$, expressed as eta-quotients. We prove that the sequence $σ_\ell^{\mathrm{mex}}(n)$ is weakly increasing for every $\ell$, and we obtain Hardy--Ramanujan type asymptotic formulas for $σ_2^{\mathrm{mex}}(n)$ and $σ_3^{\mathrm{mex}}(n)$ as $n \to \infty$. Additionally, we completely determine the parity of $σ_2^{\mathrm{mex}}(n)$, showing that it is odd precisely at generalized pentagonal numbers, and we use this characterization to compute the density of $n$ for which $σ_2^{\mathrm{mex}}(n)$ is odd versus even. Finally, for $σ_3^{\mathrm{mex}}(n)$, we prove that it is odd infinitely often, and we establish a quantitative lower bound, showing that the number of such $n$ up to $X$ grows at least like $\log \log X$.