Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Aubrey Blecher, Charlotte Brennan, Toufik Mansour

Research output: Contribution to journalArticlepeer-review

Abstract

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i. e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of "valleys" and "peaks". From this, we find the total number of "peaks" and "valleys" in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.

Original languageEnglish
Pages (from-to)788-796
Number of pages9
JournalCentral European Journal of Mathematics
Volume10
Issue number2
DOIs
StatePublished - Apr 2012

Keywords

  • Compositions
  • Generating functions
  • Partitions

ASJC Scopus subject areas

  • Mathematics (all)

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