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On the Distribution of the spt-Crank

Author

Listed:
  • 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)

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.

Suggested Citation

  • George E. Andrews & Freeman J. Dyson & Robert C. Rhoades, 2013. "On the Distribution of the spt-Crank," Mathematics, MDPI, vol. 1(3), pages 1-13, June.
    • Handle: RePEc:gam:jmathe:v:1:y:2013:i:3:p:76-88:d:26784
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