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Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

Aval, Jean-Christophe and D'Adderio, Michele and Dukes, Mark and Hicks, Angela and Le Borgne, Yvan (2014) Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers. Journal of Combinatorial Theory Series A, 123 (1). pp. 271-286. ISSN 0097-3165

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We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. 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 the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.