Authors: Jean-Christophe Aval ¹; Michele D'Adderio ; Mark Dukes ²; 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.