### 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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*143*(10), 4283-4289. https://doi.org/10.1090/S0002-9939-2015-12591-9

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*Proceedings of the American Mathematical Society*, vol. 143, no. 10, pp. 4283-4289. https://doi.org/10.1090/S0002-9939-2015-12591-9

**Partitions with fixed differences between largest and smallest parts.** / Andrews, George E.; Beck, Matthias; Robbins, Neville.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Partitions with fixed differences between largest and smallest parts

AU - Andrews, George E.

AU - Beck, Matthias

AU - Robbins, Neville

PY - 2015/10/1

Y1 - 2015/10/1

N2 - 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 Pt(q) := ∑n≥1 p(n, t) qn. Somewhat surprisingly, Pt(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.

AB - 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 Pt(q) := ∑n≥1 p(n, t) qn. Somewhat surprisingly, Pt(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.

UR - http://www.scopus.com/inward/record.url?scp=84938263648&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84938263648&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2015-12591-9

DO - 10.1090/S0002-9939-2015-12591-9

M3 - Article

AN - SCOPUS:84938263648

VL - 143

SP - 4283

EP - 4289

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 10

ER -