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

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 [4]. 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.