### 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.

Original language | English |
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Title of host publication | DMTCS Proceedings |

Subtitle of host publication | 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) |

Place of Publication | Nancy, France |

Pages | 623-634 |

Number of pages | 12 |

Publication status | Published - 2013 |

### Keywords

- q, t-Narayana
- rectangular polyominoes
- parking functions

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## Cite this

Aval, J-C., D'Adderio, M., Dukes, M., Hicks, A., & Le Borgne, Y. (2013). A q,t-analogue of Narayana numbers. In

*DMTCS Proceedings : 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)*(pp. 623-634).