Weyl group q-Kreweras numbers and cyclic sieving

Victor Reiner, Eric Sommers

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A,B,C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A,B,C,D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

Original languageEnglish (US)
Pages (from-to)819-874
Number of pages56
JournalAnnals of Combinatorics
Volume22
Issue number4
DOIs
StatePublished - Jan 1 2018

Keywords

  • Catalan number
  • Cyclic sieving phenomenon
  • Kreweras number
  • Narayana number
  • Nilpotent orbit
  • Reflection
  • Weyl group

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