on the distribution of the spt-crank
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ID: 193285
2013
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Abstract
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 “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.
| Reference Key |
rhoades2013mathematicson
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|---|---|
| Authors | ;Robert C. Rhoades;Freeman J. Dyson;George E. Andrews |
| Journal | Turkish journal of pharmaceutical sciences |
| Year | 2013 |
| DOI |
10.3390/math1030076
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| URL | |
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