Counting permutations by the number of successions within cycles

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


In this note, we consider the problem of counting (cycle) successions, i.e., occurrences of adjacent consecutive elements within cycles, of a permutation expressed in the standard form. We find an explicit formula for the number of permutations having a prescribed number of cycles and cycle successions, providing both algebraic and combinatorial proofs. As an application of our ideas, it is possible to obtain explicit formulas for the joint distribution on Sn for the statistics recording the number of cycles and adjacencies of the form j,j+d where d>0 which extends earlier results.

Original languageEnglish
Pages (from-to)1368-1376
Number of pages9
JournalDiscrete Mathematics
Issue number4
StatePublished - 6 Apr 2016

Bibliographical note

Publisher Copyright:
© 2015 Elsevier B.V. All rights reserved.


  • Combinatorial proof
  • Permutation
  • Stirling number of first kind
  • Succession

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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