Overpartitions with repeated smallest non-overlined part

TL;DR

This work extends the spt-type statistics to overpartitions by defining $\overline{\mathrm{Spt}}_k(n)$ and related restricted variants, and derives generating-function identities that express these counts as linear combinations of standard $q$-Pochhammer factors with polynomial coefficients. It establishes closed forms and recurrences for the accompanying rational functions $\overline{P}_k(q)$, $\overline{V}_k(q)$, $\overline{W}_k(q)$, and $\overline{T}_k(q)$, including parity- and congruence-restricted cases. The paper also develops restricted-parameter analogues $\overline{\mathrm{Spt}}k_o(n)$ and $\overline{\mathrm{Spt}}k_o'(n)$ with analogous recurrences and closed forms, and demonstrates how these spt-type counts decompose into linear combinations of overpartition subclass counts such as $\overline{p}(n)$, $\overline{p}_e(n)$, and $\overline{p}_o(n)$. The results yield concrete identities and corollaries relating overpartition statistics to parity/restriction counts and open avenues for combinatorial proofs of these relations.

Abstract

Inspired by Andrews' and Bachraoui's work on partitions with repeated smallest part, we extend the concept to overpartitions.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• proof