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

Nayandeep Deka Baruah, Haijun Li, Pankaj Jyoti Mahanta

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…