On the Distribution of the spt-Crank
AbstractAndrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence is unimodal, where 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.
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Andrews, G.E.; Dyson, F.J.; Rhoades, R.C. On the Distribution of the spt-Crank. Mathematics 2013, 1, 76-88.
Andrews GE, Dyson FJ, Rhoades RC. On the Distribution of the spt-Crank. Mathematics. 2013; 1(3):76-88.Chicago/Turabian Style
Andrews, George E.; Dyson, Freeman J.; Rhoades, Robert C. 2013. "On the Distribution of the spt-Crank." Mathematics 1, no. 3: 76-88.