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

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.

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