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

We study the statistics areaarea, bouncebounce and dinvdinv on the set of parallelogram polyominoes having a rectangular m times n   bounding box. We show that the bi-statistics (area,bounce)(area,bounce) and (area,dinv)(area,dinv) give rise to the same q,tq,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,tq,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,tq,t-Narayana polynomials which relates them to the famous diagonal harmonics.