On the distribution of the spt-crank

George E. Andrews, Freeman J. Dyson, Robert C. Rhoades

Research output: Contribution to journalArticle

11 Scopus citations

Abstract

Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence fNS(m; n)gm 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.

Original languageEnglish (US)
Pages (from-to)76-88
Number of pages13
JournalMathematics
Volume1
Issue number3
DOIs
StatePublished - Sep 1 2013

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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