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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
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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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