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Mathematics 2013, 1(3), 76-88; doi:10.3390/math1030076

On the Distribution of the spt-Crank

1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA 2 Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, NJ 08540, USA 3 Stanford University, Department of Mathematics, Bldg 380, Stanford, CA 94305, USA
* Author to whom correspondence should be addressed.
Received: 16 February 2013 / Revised: 10 April 2013 / Accepted: 10 April 2013 / Published: 28 June 2013
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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 partitions; partition crank; partition rank; spt-crank; unimodal
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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.

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