Jean-Christophe Aval ; Michele D'Adderio ; Mark Dukes ; Angela Hicks ; Yvan Le Borgne - A $q,t-$analogue of Narayana numbers

dmtcs:2329 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2329
A $q,t-$analogue of Narayana numbersArticle

Authors: Jean-Christophe Aval 1; Michele D'Adderio ; Mark Dukes ORCID2; Angela Hicks 3; Yvan Le Borgne 1

  • 1 Laboratoire Bordelais de Recherche en Informatique
  • 2 Department of Computer and Information Sciences [Univ Strathclyde]
  • 3 Department of Mathematics [Univ California San Diego]

We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbers, which was introduced by two of these authors in a recent paper. We prove the main conjectures of that same work, i.e. the symmetries in $q$ and $t$, and in $m$ and $n$ of these polynomials, by providing a symmetric functions interpretation which relates them to the famous diagonal harmonics.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: q,t-Narayana,rectangular polyominoes,parking functions.,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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