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

  • Authors:

  • Jean-Christophe Aval;Michele Dadderio;Mark Dukes;Angela Hicks;Yvan Le Borgne

  • Affiliations:

  • LaBRI, Université de Bordeaux, CNRS, 351 cours de la Libération, 33405 Talence, France;Université Libre de Bruxelles (ULB), Département de Mathématique, Boulevard du Triomphe, B-1050 Bruxelles, Belgium;University of Strathclyde, Department of Computer and Information Sciences, 16 Richmond Street, Glasgow G1 1XQ, Scotland, United Kingdom;Stanford University, Department of Mathematics, building 380, Stanford, CA 94305, USA;LaBRI, Université de Bordeaux, CNRS, 351 cours de la Libération, 33405 Talence, France

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2014

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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 [4]. 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.