The spt-crank for overpartitions (Q2928542)

For \(i=1\) or \(2\) let \(\overline{\text{spt}}_i(n)\) denote half of the number of smallest parts in the overpartitions of \(n\) whose smallest part is congruent to \(i\) modulo \(2\). Five congruences for \(\text{spt}_i(n)\) modulo \(3\) and \(5\) were found in [\textit{K. Bringmann} et al., J. Number Theory 129, No. 7, 1758--1772 (2009; Zbl 1248.11076)]. For example, we have NEWLINE\[NEWLINE \text{spt}_2(3n+1) \equiv 0 \pmod{3} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \text{spt}_1(5n) \equiv 0 \pmod{5}. NEWLINE\]NEWLINE In the paper under review the authors give combinatorial decompositions of these congruences using ``marked overpartitions'' and a statistic called the \(\overline{\text{sptcrank}}\). For example, they show that the residue of the \(\overline{\text{sptcrank}}\) modulo \(5\) divides the marked overpartitions of \(5n\) with smallest part odd into five equal classes. This corresponds to the second congruence above, in the sense that there is a weight-preserving bijective mapping from marked overpartitions to certain vector partitions whose two-variable generating function simplifies at \(z=1\) to the smallest parts generating function.NEWLINENEWLINEThe authors also study congruences for the number of smallest parts in the partitions of \(n\) without repeated odd parts and with smallest part even. They prove three such congruences and give combinatorial decompositions in terms of weighted vector partitions. The role of ``marked'' objects is uncertain in this case and the authors leave this open.NEWLINENEWLINEFinally, they determine the parity of \(\text{spt}_i(n)\). They show that \(\text{spt}_1(n)\) (resp. \(\text{spt}_2(n)\)) is odd if and only if \(n\) is a twice a square or an odd (resp. even) square.