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On the Distribution of the spt-Crank
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, NJ 08540, USA
Stanford University, Department of Mathematics, Bldg 380, Stanford, CA 94305, USA
* Author to whom correspondence should be addressed.
Received: 16 February 2013; in revised form: 10 April 2013 / Accepted: 10 April 2013 / Published: 28 June 2013
Abstract: Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence is unimodal, where 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
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MDPI and ACS Style
Andrews, G.E.; Dyson, F.J.; Rhoades, R.C. On the Distribution of the spt-Crank. Mathematics 2013, 1, 76-88.
Andrews GE, Dyson FJ, Rhoades RC. On the Distribution of the spt-Crank. Mathematics. 2013; 1(3):76-88.
Andrews, George E.; Dyson, Freeman J.; Rhoades, Robert C. 2013. "On the Distribution of the spt-Crank." Mathematics 1, no. 3: 76-88.