Geodesic
A $q,t-$analogue of Narayana numbers
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

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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. 
@article{DMTCS_2013_special_264_a13,
     author = {Aval, Jean-Christophe and D'Adderio, Michele and Dukes, Mark and Hicks, Angela and Le Borgne, Yvan},
     title = {A $q,t-$analogue of {Narayana} numbers},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)},
     year = {2013},
     doi = {10.46298/dmtcs.2329},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2329/}
}
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Aval, Jean-Christophe; D'Adderio, Michele; Dukes, Mark; Hicks, Angela; Le Borgne, Yvan. A $q,t-$analogue of Narayana numbers. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013). doi : 10.46298/dmtcs.2329. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2329/

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