Euler's partition theorem with upper bounds on multiplicities

William Y.C. Chen, Ae Ja Yee, Albert J.W. Zhu

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We show that the number of partitions of n with alternating sum κ such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's partition theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a generalization of Euler's partition theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of n with alternating sum κ such that the multiplicity of each even part is bounded by 2m + 1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is also bounded by 2m+1. We provide a combinatorial proof as well.

Original languageEnglish (US)
Pages (from-to)1-15
Number of pages15
JournalElectronic Journal of Combinatorics
Volume19
Issue number3
DOIs
StatePublished - Oct 4 2012

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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