Partitions with fixed differences between largest and smallest parts

George E. Andrews; Matthias Beck; Neville Robbins

doi:10.1090/S0002-9939-2015-12591-9

Partitions with fixed differences between largest and smallest parts

George E. Andrews
, Matthias Beck
, Neville Robbins

Mathematics

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

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) qⁿ. 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)
Pages (from-to)	4283-4289
Number of pages	7
Journal	Proceedings of the American Mathematical Society
Volume	143
Issue number	10
DOIs	https://doi.org/10.1090/S0002-9939-2015-12591-9
State	Published - Oct 1 2015

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1090/S0002-9939-2015-12591-9

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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 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.",

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AU - Andrews, George E.

AU - Beck, Matthias

AU - Robbins, Neville

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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.

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

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JF - Proceedings of the American Mathematical Society

IS - 10

ER -

Partitions with fixed differences between largest and smallest parts

Abstract

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