New refined enumerations of set partitions related to sorting

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


In this paper, we consider a statistic defined on the sequential representation of a finite set partition which tracks the number of steps required, i.e. pushes, to put the sequence in weakly increasing order. A generating function is computed for the distribution of the statistic that records the number of pushes considered jointly with some other combinatorial parameters on the set of partitions of [n] = {1, 2, . . . , n} for each fixed n. Some further attention is paid to the distribution for the number of pushes by itself. Generating function formulas are also found for the comparable distribution on the set of non-crossing and non-nesting partitions of [n]. As a consequence of our results, one obtains new polynomial generalizations of the Stirling, Bell and Catalan numbers.

Original languageEnglish
Pages (from-to)1588-1603
Number of pages16
JournalJournal of Difference Equations and Applications
Issue number10
StatePublished - 3 Oct 2018

Bibliographical note

Publisher Copyright:
© 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.


  • 05A15
  • 05A18
  • Generating functions
  • combinatorial statistic
  • finite set partitions
  • non-crossing partitions

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics


Dive into the research topics of 'New refined enumerations of set partitions related to sorting'. Together they form a unique fingerprint.

Cite this