### Abstract

We study the number p(n, t) of partitions of n with difference t between largest and smallest parts. Our main result is an explicit formula for the generating function P_{t}(q) := ∑_{n}≥1 p(n, t) q^{n}. Somewhat surprisingly, P_{t}(q) is a rational function for t > 1; equivalently, p(n, t) is a quasipolynomial in n for fixed t > 1. Our result generalizes to partitions with an arbitrary number of specified distances.

Original language | English (US) |
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Pages (from-to) | 4283-4289 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 143 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2015 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Andrews, G. E., Beck, M., & Robbins, N. (2015). Partitions with fixed differences between largest and smallest parts.

*Proceedings of the American Mathematical Society*,*143*(10), 4283-4289. https://doi.org/10.1090/S0002-9939-2015-12591-9