On the Distribution of the spt -Crank

Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence {N-S(m; n)}(m) is unimodal, where N-S(m; n) is the number of S-partitions of size n with crank m weight by the spt -crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conjecture is true and allows us to asymptotically establish the conjecture. Additionally, we give an asymptotic study for the distribution of the spt -crank statistic. Finally, we give some speculations about a definition for the spt - crank in terms of "marked" partitions. A "marked" partition is an unrestricted integer partition where each part is marked with a multiplicity number. It remains an interesting and apparently challenging problem to interpret the spt - crank in terms of ordinary integer partitions.