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On the Class of Dominant and Subordinate Products

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.
Mathematics 2013, 1(3), 76-88;
Received: 16 February 2013 / Revised: 10 April 2013 / Accepted: 10 April 2013 / Published: 28 June 2013
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. View Full-Text
Keywords: partitions; partition crank; partition rank; spt-crank; unimodal partitions; partition crank; partition rank; spt-crank; unimodal
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.

AMA Style

Andrews GE, Dyson FJ, Rhoades RC. On the Distribution of the spt-Crank. Mathematics. 2013; 1(3):76-88.

Chicago/Turabian Style

Andrews, George E., Freeman J. Dyson, and Robert C. Rhoades 2013. "On the Distribution of the spt-Crank" Mathematics 1, no. 3: 76-88.

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