On the sum of odd minimal excludants over overpartitions

Source abstract

Andrews and Newman introduced the minimal excludant $\mathrm{mex}(λ)$ of an integer partition $λ$ and studied the summatory function $σ\mathrm{mex}(n)$, and Baruah et al. refined this to the odd- and even-restricted functions $σ_o\mathrm{mex}(n)$ and $σ_e\mathrm{mex}(n)$. In this paper, we introduce and study the overpartition analogue $\overline{σ_o\mathrm{mex}}(n)$, defined as the sum of odd minimal excludants over all overpartitions of $n$. We first derive the exact generating function for $\overline{σ_o\mathrm{mex}}(n)$, 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 $\varphi(q)^2$ verified via the Gordon--Hughes--Newman--Ligozat criterion, we establish an infinite family of congruences satisfied by $\overline{σ_o\mathrm{mex}}(n)$. Consequently, we obtain that $\overline{σ_o\mathrm{mex}}(0)=1$ and, for $n\ge1$, $\overline{σ_o\mathrm{mex}}(n) \equiv 0 \pmod{4}$ if and only if $n$ is a perfect square. We further obtain congruences for the partial sums and self-convolution of $\overline{σ_o\mathrm{mex}}(n)$; 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 $\overline{σ_o\mathrm{mex}}(n)$ via the Wright circle method.