Counting peaks and valleys in a partition of a set

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


A partition π of the set [n] = {1,2,...,n} is a collection {B1,B2,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. In this paper, we find an explicit formula for the generating function for the number of partitions of [n] with exactly k blocks according to the number of peaks (valleys) in terms of Chebyshev polynomials of the second kind. Furthermore, we calculate explicit formulas for the total number of peaks and valleys in all the partitions of [n] with exactly k blocks, providing both algebraic and combinatorial proofs.

Original languageEnglish
Pages (from-to)1-16
Number of pages16
JournalJournal of Integer Sequences
Issue number6
StatePublished - 2010


  • Generating function
  • Peak
  • Recurrence relation
  • Set partition
  • Valley

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


Dive into the research topics of 'Counting peaks and valleys in a partition of a set'. Together they form a unique fingerprint.

Cite this