Parity successions in set partitions

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


Suppose that the elements within each block of a partition π of [n]={1,2,.,n} are written in ascending order. By a parity succession, we will mean a pair of adjacent elements x and y within some block of π such that x≡y(mod2). Here, we consider the problem of counting the partitions of [n] according to the number of successions, extending recent results concerning successions on subsets and permutations. Using linear algebra, we determine a formula for the generating function which counts partitions having a fixed number of blocks according to size and number of successions. Furthermore, a special case of our formula yields an explicit recurrence for the generating function which counts the parity-alternating partitions of [n], i.e., those that contain no successions.

Original languageEnglish
Pages (from-to)2642-2650
Number of pages9
JournalLinear Algebra and Its Applications
Issue number9
StatePublished - 1 Nov 2013


  • Generating function
  • Parity succession
  • Set partition
  • Tridiagonal matrix

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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