Combinatorial parameters on bargraphs of permutations

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


In this paper, we consider statistics on permutations of length n represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we find the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new refined count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which S n is a subset. In addition to finding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit expressions for the sign balance of these statistics, from which the comparable results on permutations follow as special cases.

Original languageEnglish
Pages (from-to)1-16
Number of pages16
JournalTransactions on Combinatorics
Issue number2
StatePublished - 1 Jun 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018 University of Isfahan.


  • Bargraph
  • Combinatorial statistic
  • Permutations
  • Q-generalization

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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