Statistics on bargraphs of inversion sequences of permutations

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


An inversion sequence (x1, …, xn) is one such that 1 ≤ xi ≤ i for all 1 ≤ i ≤ n. We first consider the joint distribution of the area and perimeter statistics on the set In of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived from recurrences satisfied by this distri-bution. Explicit formulas for the generating functions are found in some special cases as are expressions for the totals of the respective statistics on In. A similar treatment is provided for the joint distribution on In for the statistics recording the number of levels, descents and ascents. Some connections are made between specific cases of this latter distribution and the Stirling numbers of the first kind and Eulerian numbers.

Original languageEnglish
Pages (from-to)42-49
Number of pages8
JournalDiscrete Mathematics Letters
StatePublished - 2020

Bibliographical note

Publisher Copyright:
© 2020 the authors.


  • Bargraph
  • Combinatorial statistic
  • Inversion sequence

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


Dive into the research topics of 'Statistics on bargraphs of inversion sequences of permutations'. Together they form a unique fingerprint.

Cite this