Another SPT crank for the number of smallest parts in overpartitions with even smallest part

By using the M2-rank of an overpartition as well as a residual crank, we give another combinatorial refinement of the congruences spt¯2(3n)≡spt¯2(3n+1)≡0(mod3). Here spt¯2(n) is the total number of appearances 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 M2-rank of an overpartition. This congruence was previously refined using the rank of an overpartition.