On the Distribution of the spt-Crank

George E. Andrews, Freeman J. Dyson and Robert C. Rhoades
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George E. Andrews: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
Freeman J. Dyson: Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, NJ 08540, USA
Robert C. Rhoades: Stanford University, Department of Mathematics, Bldg 380, Stanford, CA 94305, USA

Mathematics, 2013, vol. 1, issue 3, 1-13

Abstract: 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.

Keywords: partitions; partition crank; partition rank; spt-crank; unimodal (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2013
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