Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

Jean-Christophe Aval, Michele D'Adderio, Mark Dukes, Angela Hicks, Yvan Le Borgne

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Abstract

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.
Original languageEnglish
Pages (from-to)271-286
Number of pages16
JournalJournal of Combinatorial Theory Series A
Volume123
Issue number1
Early online date24 Jan 2014
DOIs
Publication statusPublished - Apr 2014

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Keywords

  • q,tq,t-Narayana
  • parallelogram polyominoes
  • parking functions

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