Block diagonally symmetric lozenge tilings
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908283585953792 |
|---|---|
| author | Byun, Seok Hyun Lee, Yi-Lin |
| author_facet | Byun, Seok Hyun Lee, Yi-Lin |
| contents | We introduce a new symmetry class of both boxed plane partitions and lozenge tilings of a hexagon, called the $\mathbf{r}$-block diagonal symmetry class, where $\mathbf{r}$ is an $n$-tuple of non-negative integers. We prove that the tiling generating function of this symmetry class under a certain weight assignment is given by a simple product formula. As a consequence, the volume generating function of $\mathbf{r}$-block symmetric plane partitions is obtained. Additionally, we consider $(\mathbf{r},\mathbf{r^{\prime}})$-block diagonally symmetric lozenge tilings by embedding the hexagon into a cylinder and present an identity for the signed enumeration of this symmetry class in specific cases. Two methods are provided to study this symmetry class: (1) the method of non-intersecting lattice paths with a modification, and (2) interpreting weighted lozenge tilings algebraically as (skew) Schur polynomials and applying the dual Pieri rule. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19249 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Block diagonally symmetric lozenge tilings Byun, Seok Hyun Lee, Yi-Lin Combinatorics 05A15, 05A19, 05B45, 05E05 We introduce a new symmetry class of both boxed plane partitions and lozenge tilings of a hexagon, called the $\mathbf{r}$-block diagonal symmetry class, where $\mathbf{r}$ is an $n$-tuple of non-negative integers. We prove that the tiling generating function of this symmetry class under a certain weight assignment is given by a simple product formula. As a consequence, the volume generating function of $\mathbf{r}$-block symmetric plane partitions is obtained. Additionally, we consider $(\mathbf{r},\mathbf{r^{\prime}})$-block diagonally symmetric lozenge tilings by embedding the hexagon into a cylinder and present an identity for the signed enumeration of this symmetry class in specific cases. Two methods are provided to study this symmetry class: (1) the method of non-intersecting lattice paths with a modification, and (2) interpreting weighted lozenge tilings algebraically as (skew) Schur polynomials and applying the dual Pieri rule. |
| title | Block diagonally symmetric lozenge tilings |
| topic | Combinatorics 05A15, 05A19, 05B45, 05E05 |
| url | https://arxiv.org/abs/2503.19249 |