Another proof of two modulo 3 congruences and another SPT crank for the number of smallest parts in overpartitions with even smallest part

TL;DR

This paper provides a new combinatorial proof of certain modulo 3 congruences related to the count of smallest parts in specific overpartitions, using the $M_2$-rank and residual crank concepts.

Contribution

It introduces a novel combinatorial refinement of overpartition smallest part congruences using the $M_2$-rank and residual crank, extending previous rank-based refinements.

Findings

Proves modulo 3 congruences for overpartition smallest parts.

Uses Bailey's Lemma and rank difference formulas in the proof.

Provides a new combinatorial perspective on overpartition congruences.

Abstract

By considering the $M_2$-rank of an overpartition as well as a residual crank, we give another combinatorial refinement of the congruences $\overline{\mbox{spt}}_2(3n)\equiv \overline{\mbox{spt}}_2(3n+1)\equiv 0\pmod{3}$. Here $\overline{\mbox{spt}}_2(n)$ is the total number of occurrences of the smallest parts among the overpartitions of $n$ where the smallest part is even and not overlined. Our proof depends on Bailey's Lemma and the rank difference formulas of Lovejoy and Osburn for the $M_2$-rank of an overpartition. This congruence, along with a modulo $5$ congruence, has previously been refined using the rank of an overpartition.