Counting water cells in bargraphs of compositions and set partitions

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


In this paper, we consider statistics on compositions and set partitions represented geometrically as bargraphs. By a water cell, we mean a unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within the area subtended by the bargraph. That is, if a large amount of a liquid were poured onto the bargraph from above and allowed to drain freely, then the water cells are precisely those cells where the liquid would collect. In this paper, we count both compositions and set partitions according to the number of descents and water cells in their bargraph representations and determine generating function formulas for the joint distributions on the respective structures. Comparable generating functions that count non-crossing and non-nesting partitions are also found. Finally, we determine explicit formulas for the sign balance and for the first moment of the water cell statistic on set partitions, providing both algebraic and combinatorial proofs.

Original languageEnglish
Pages (from-to)413-438
Number of pages26
JournalApplicable Analysis and Discrete Mathematics
Issue number2
StatePublished - 1 Oct 2018

Bibliographical note

Publisher Copyright:
© 2018, University of Belgrade.


  • Bargraphs
  • Compositions
  • Polynomial generalization
  • Set partitions

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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