### 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 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Fingerprint Dive into the research topics of 'On the distribution of the spt-crank'. Together they form a unique fingerprint.

## Cite this

*Mathematics*,

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