### 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 language | English (US) |
---|---|

Pages (from-to) | 76-88 |

Number of pages | 13 |

Journal | Mathematics |

Volume | 1 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematics*,

*1*(3), 76-88. https://doi.org/10.3390/math1030076

}

*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.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the distribution of the spt-crank

AU - Andrews, George E.

AU - Dyson, Freeman J.

AU - Rhoades, Robert C.

PY - 2013/9/1

Y1 - 2013/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84884352247&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884352247&partnerID=8YFLogxK

U2 - 10.3390/math1030076

DO - 10.3390/math1030076

M3 - Article

AN - SCOPUS:84884352247

VL - 1

SP - 76

EP - 88

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 3

ER -