Compositions with restricted parts

Research output: Contribution to journalArticle

Abstract

Euler showed that the number of partitions of n into distinct parts equals the number of partitions of n into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang. Analogous to Euler's partition theorem, it is known that the number of compositions of n with odd parts equals the number of compositions of n+1 with parts greater than one, as both numbers equal the Fibonacci number Fn. Recently, Sills provided a bijective proof for this result using binary sequences, and Munagi proved a generalization similar to Glaisher's result using the zigzag graphs of compositions. Extending Sills’ bijection, we obtain a further generalization which is analogous to Franklin's result. We establish, both analytically and combinatorially, two closed formulae for the number of compositions with restricted parts appearing in our generalization. We also prove some composition analogues for the conjectures of Beck.

Original languageEnglish (US)
Article number111875
JournalDiscrete Mathematics
Volume343
Issue number7
DOIs
StatePublished - Jul 2020

Keywords

  • Composition
  • Euler's partition theorem
  • Restricted parts

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Fingerprint Dive into the research topics of 'Compositions with restricted parts'. Together they form a unique fingerprint.

  • Cite this