Domino Tilings, Domino Shuffling, and the Nabla Operator
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913669926879232 |
|---|---|
| author | Cavey, Ian Lee, Yi-Lin |
| author_facet | Cavey, Ian Lee, Yi-Lin |
| contents | We study domino tilings of certain regions $R_λ$, indexed by partitions $λ$, weighted according to generalized area and dinv statistics. These statistics arise from the $q,t$-Catalan combinatorics and Macdonald polynomials. We present a formula for the generating polynomial of these domino tilings in terms of the Bergeron--Garsia nabla operator. When $λ= (n^n)$ is a square shape, domino tilings of $R_λ$ are equivalent to those of the Aztec diamond of order $n$. In this case, we give a new product formula for the resulting polynomials by domino shuffling and its connection with alternating sign matrices. In particular, we obtain a combinatorial proof of the joint symmetry of the generalized area and dinv statistics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17765 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Domino Tilings, Domino Shuffling, and the Nabla Operator Cavey, Ian Lee, Yi-Lin Combinatorics 05A15, 05A19, 05B45, 05E05 We study domino tilings of certain regions $R_λ$, indexed by partitions $λ$, weighted according to generalized area and dinv statistics. These statistics arise from the $q,t$-Catalan combinatorics and Macdonald polynomials. We present a formula for the generating polynomial of these domino tilings in terms of the Bergeron--Garsia nabla operator. When $λ= (n^n)$ is a square shape, domino tilings of $R_λ$ are equivalent to those of the Aztec diamond of order $n$. In this case, we give a new product formula for the resulting polynomials by domino shuffling and its connection with alternating sign matrices. In particular, we obtain a combinatorial proof of the joint symmetry of the generalized area and dinv statistics. |
| title | Domino Tilings, Domino Shuffling, and the Nabla Operator |
| topic | Combinatorics 05A15, 05A19, 05B45, 05E05 |
| url | https://arxiv.org/abs/2501.17765 |