# A $q,t-$analogue of Narayana numbers

Abstract : 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.
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Communication dans un congrès
Alain Goupil and Gilles Schaeffer. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp.623-634, 2013, DMTCS Proceedings

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https://hal.inria.fr/hal-01229669
Contributeur : Alain Monteil <>
Soumis le : mardi 17 novembre 2015 - 10:19:38
Dernière modification le : jeudi 11 janvier 2018 - 06:20:17
Document(s) archivé(s) le : jeudi 18 février 2016 - 11:34:25

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• HAL Id : hal-01229669, version 1

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Jean-Christophe Aval, Michele D'Adderio, Mark Dukes, Angela Hicks, Yvan Le Borgne. A $q,t-$analogue of Narayana numbers. Alain Goupil and Gilles Schaeffer. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp.623-634, 2013, DMTCS Proceedings. 〈hal-01229669〉

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