%0 Journal Article
%T On the Distribution of the spt-Crank
%A George E. Andrews
%A Freeman J. Dyson
%A Robert C. Rhoades
%J Mathematics
%D 2013
%I MDPI AG
%R 10.3390/math1030076
%X Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence {NS ( m, n) } m is unimodal, where NS ( 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 ˇ°markedˇ± partitions. A ˇ°markedˇ± 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.
%K partitions
%K partition crank
%K partition rank
%K spt-crank
%K unimodal
%U http://www.mdpi.com/2227-7390/1/3/76