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

On the Distribution of the spt-Crank

Received: 16 February 2013; in revised form: 10 April 2013 / Accepted: 10 April 2013 / Published: 28 June 2013
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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.

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

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