Counting permutations by cyclic peaks and valleys

Chak On Chow, Shi Mei Ma, Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


In this paper, we study the generating functions for the number of permutations having a prescribed number of cyclic peaks or valleys. We derive closed form expressions for these functions by use of various algebraic methods. When restricted to the set of derangements (i.e., fixed point free permutations), the evaluation at −1 of the generating function for the number of cyclic valleys gives the Pell number. We provide a bijective proof of this result, which can be extended to the entire symmetric group.

Original languageEnglish
Pages (from-to)43-54
Number of pages12
JournalAnnales Mathematicae et Informaticae
StatePublished - 2014

Bibliographical note

Funding Information:
Supported by grants of the Fondo de Investigaciones Sanitarias (FIS) (96/1409), of the Dirección General de Enseñanza Superior, MEC (PB95-0035), and of the Fundación de la Sociedad Española de Alergia.

Publisher Copyright:
© 2014, Eszterhazy Karoly College. All rights reserved.


  • Cyclic valleys
  • Derangements
  • Involutions
  • Pell numbers

ASJC Scopus subject areas

  • Computer Science (all)
  • Mathematics (all)


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