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.
- parallelogram polyominoes
- parking functions
Aval, J-C., D'Adderio, M., Dukes, M., Hicks, A., & Le Borgne, Y. (2014). Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers. Journal of Combinatorial Theory Series A , 123(1), 271-286. https://doi.org/10.1016/j.jcta.2013.09.001