Counting permutations by the number of vertical edges in their bargraphs

Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


In this paper, we consider a new statistic on the set Sn of permutations of length n which records the number of vertical edges within their bargraph representations. In-deed, we determine the distribution of this statistic on a class of multi-permutations having Sn as a subset. We compute an explicit formula for the total number of vertical edges within all members of this class and also for the number of interior vertices in the corresponding bargraphs. A generating function formula is determined and some closely related statistics are considered. Finally, we show that the total number of vertical edges in all the members of Sn equals the number of runs in the members of Sn+1 by constructing an explicit bijection which applies more generally to multi-permutations.

Original languageAmerican English
Pages (from-to)279-290
Number of pages12
JournalJournal of Automata, Languages and Combinatorics
Issue number4
StatePublished - 2020


  • Bargraphs
  • Generating functions
  • Multi-set permutations
  • Permutation statistics

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


Dive into the research topics of 'Counting permutations by the number of vertical edges in their bargraphs'. Together they form a unique fingerprint.

Cite this