A q,t-analogue of Narayana numbers

Aval, Jean-Christophe and D'Adderio, Michele and Dukes, Mark and Hicks, Angela and Le Borgne, Yvan; (2013) A q,t-analogue of Narayana numbers. In: DMTCS Proceedings. Discrete Mathematics & Theoretical Computer Science, Nancy, France, pp. 623-634. (http://www.dmtcs.org/dmtcs-ojs/index.php/proceedin...)

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Abstract

We study the statistics area, bounce and dinv associated to polyominoes in a rectangular box m times n. We show that the bi-statistics (area, bounce) and (area, 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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