On the sum of odd minimal excludants over overpartitions

Abstract

Andrews and Newman introduced the minimal excludant $\mathrm{mex}(\lambda )$ of an integer partition $\lambda$ and studied the summatory function $\sigma \mathrm{mex}(n)$, and Baruah et al. refined this to the odd- and even-restricted functions ${\sigma }_o\mathrm{mex}(n)$ and ${\sigma }_e\mathrm{mex}(n)$. In this paper, we introduce and study the overpartition analogue $\overline{{\sigma }_o\mathrm{mex}}(n)$, defined as the sum of odd minimal excludants over all overpartitions of $n$. We first derive the exact generating function for $\overline{{\sigma }_o\mathrm{mex}}(n)$, and relate it to the bivariate generating function of Aricheta and Donato for the overpartition minimal excludant. Using elementary $q$-series arguments, together with a weight-one eta-quotient identity for $\phi (q{)}^2$ verified via the Gordon--Hughes--Newman--Ligozat criterion, we establish an infinite family of congruences satisfied by $\overline{{\sigma }_o\mathrm{mex}}(n)$. Consequently, we obtain that $\overline{{\sigma }_o\mathrm{mex}}(0)=1$ and, for $n\ge 1$, $\overline{{\sigma }_o\mathrm{mex}}(n)\equiv 0(\mathrm{mod}4)$ if and only if $n$ is a perfect square. We further obtain congruences for the partial sums and self-convolution of $\overline{{\sigma }_o\mathrm{mex}}(n)$; in particular, an infinite family of congruences modulo $8$ for the self-convolution, expressed in terms of the divisor functions $d_1(n)$ and $d_3(n)$. We conclude the paper by establishing a Hardy--Ramanujan-type asymptotic formula for $\overline{{\sigma }_o\mathrm{mex}}(n)$ via the Wright circle method.