Refinements of Beck-type partition identities

T. Amdeberhan, G. E. Andrews, C. Ballantine

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Franklin's identity generalizes Euler's identity and states that the number of partitions of n with j different parts divisible by r equals the number of partitions of n with j different parts repeated at least r times. In this article, we give a refinement of Franklin's identity when j=1. We prove Franklin's identity when j=1, r=2 for partitions with fixed perimeter, i.e., fixed largest hook. We also derive a Beck-type identity for partitions with fixed perimeter: the excess in the number of parts in all partitions into odd parts with perimeter M over the number of parts in all partitions into distinct parts with perimeter M equals the number of partitions with perimeter M whose set of even parts is a singleton. We provide analytic and combinatorial proofs of our results.

Original languageEnglish (US)
Article number113110
JournalDiscrete Mathematics
Volume345
Issue number12
DOIs
StatePublished - Dec 2022

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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