A $q,t-$analogue of Narayana numbers
We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbe...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2013-01-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2329/pdf |
| Summary: | We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) 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. |
|---|---|
| ISSN: | 1365-8050 |