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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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Jean-Christophe Aval, Michele d'Adderio, Mark Dukes, Angela Hicks, yvan Le Borgne. A $q,t-$analogue of Narayana numbers. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.623-634, ⟨10.46298/dmtcs.2329⟩. ⟨hal-01229669⟩



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