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

Abstract

Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(\pi)$, appears $k$ times and every overlined part is bigger than $s(\pi)$. 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 $s(\pi)$ and all parts other than $s(\pi)$ are incongruent modulo $2$ with $s(\pi)$. 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 $s(\pi)$ is even (resp., odd), and let $$\overline{\mathrm{spt}}k_o'(n)=b_e(k,n)-b_o(k,n).$$ Recently, Malik and Sarma (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.