Euler's partition theorem and the combinatorics of ℓ-sequences

Carla D. Savage, Ae Ja Yee

Research output: Contribution to journalArticle

11 Scopus citations

Abstract

Euler's partition theorem states that the number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1. It was shown by Bousquet-Mélou and Eriksson in [M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, Ramanujan J. 1 (2) (1997) 165-185] that a similar result holds when "odd parts" is replaced by "parts that are sums of successive terms of an ℓ-sequence" and the ratio "1" is replaced by a root of the characteristic polynomial of the ℓ-sequence. This generalization of Euler's theorem is intrinsically different from the many others that have appeared, as it involves a family of partitions constrained by the ratio of successive parts. In this paper, we provide a surprisingly simple bijection for this result, a question suggested by Richard Stanley. In fact, we give a parametrized family of bijections, that include, as special cases, Sylvester's bijection and a bijection for the lecture hall theorem. We introduce Sylvester diagrams as a way to visualize these bijections and deduce their properties. In proving the bijections, we uncover the intrinsic role played by the combinatorics of ℓ-sequences and use this structure to give a combinatorial characterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described.

Original languageEnglish (US)
Pages (from-to)967-996
Number of pages30
JournalJournal of Combinatorial Theory. Series A
Volume115
Issue number6
DOIs
StatePublished - Aug 1 2008

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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