On the distribution of the spt-crank

George E. Andrews, Freeman J. Dyson, Robert C. Rhoades

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence fNS(m; n)gm is unimodal, where NS(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.

Original languageEnglish (US)
Pages (from-to)76-88
Number of pages13
JournalMathematics
Volume1
Issue number3
DOIs
StatePublished - Sep 1 2013

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Partition
Integer Partitions
M-sequence
Speculation
Statistic
Multiplicity
Heuristics

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Andrews, George E. ; Dyson, Freeman J. ; Rhoades, Robert C. / On the distribution of the spt-crank. In: Mathematics. 2013 ; Vol. 1, No. 3. pp. 76-88.
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Andrews, GE, Dyson, FJ & Rhoades, RC 2013, 'On the distribution of the spt-crank', Mathematics, vol. 1, no. 3, pp. 76-88. https://doi.org/10.3390/math1030076

On the distribution of the spt-crank. / Andrews, George E.; Dyson, Freeman J.; Rhoades, Robert C.

In: Mathematics, Vol. 1, No. 3, 01.09.2013, p. 76-88.

Research output: Contribution to journalArticle

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