Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part

TL;DR

The paper addresses combinatorial proofs of identities for overpartition statistics that involve repetition of the smallest non-overlined part. It defines and uses the families $\overline{\mathrm{spt}}k(n)$, $\overline{\mathrm{spt}}k_o(n)$, and related parity counts, extending Malik and Sharma's analytic $q$-series results to a bijective framework. The main method constructs explicit bijections (via maps $f_1$–$f_4$ and SAME/OPPOSITE nature concepts) between overpartition sets counted by the $\overline{\mathrm{spt}}$-functions and classical overpartition classes such as $\overline{P}_{ex}(n)$, $\overline{P}_e(n)$, and $\overline{P}_{oex}(n)$. These combinatorial proofs validate the $k=1$ corollaries, namely $\overline{\mathrm{spt}}1(n)+\overline{\mathrm{spt}}1(n-1)=\overline{p}_{ex}(n)$, $\overline{\mathrm{spt}}1_o(n)+\overline{\mathrm{spt}}1_o(n-2)=2\overline{p}_e(n-1)+\overline{p}_{oex}(n-1)$, and $\overline{\mathrm{spt}}1_o'(n)+\overline{\mathrm{spt}}1_o'(n-2)=-\overline{p}_{oex}'(n-1)$, providing a concrete combinatorial interpretation of the analytic identities.

Abstract

Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(π)$, appears $k$ times and every overlined part is bigger than $s(π)$. Let $\overline{\mathrm{spt}}k_o(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part appears $k$ times, every overlined part is bigger than $s(π)$ and all parts other than $s(π)$ are incongruent modulo $2$ with $s(π)$. Also, let $b_e(k,n)$ (resp., $b_o(k,n)$) denote the number of overpartitions of $n$ counted by $\overline{\mathrm{spt}}k_o(n)$ where the number of parts greater than $s(π)$ is even (resp., odd), and let $$\overline{\mathrm{spt}}k_o'(n)=b_e(k,n)-b_o(k,n).$$ Recently, Malik and Sharma (arXiv:2601.15601v1) expressed the generating functions of these partition functions in terms of linear combinations of $q$-series with polynomials in $q$ as coefficients. As corollaries, they derived some partition identities involving the functions for $k=1$ and sought for combinatorial proofs of their results. In this paper, we present some desired proofs.

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• proof : Proof of Theorem \ref{['cor4.1']}
• proof : Proof of Theorem \ref{['cor4.2']}
• proof : Proof of Theorem \ref{['cor4.3']}