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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.07984 |
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| _version_ | 1866918153337962496 |
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| author | Plante, Matthew Roby, Tom |
| author_facet | Plante, Matthew Roby, Tom |
| contents | Given a finite poset $P$, we study the _whirling_ action on vertex-labelings of $P$ with the elements $\{0,1,2,\dotsc ,k\}$. When such labelings are (weakly) order-reversing, we call them $k$-bounded $P$-partitions. We give a general equivariant bijection between $k$-bounded $P$-partitions and order ideals of the poset $P\times [k]$ which conveys whirling to the well-studied rowmotion operator. As an application, we derive periodicity and homomesy results for rowmotion acting on the chain of V's poset $V \times [k]$. We are able to generalize some of these results to the more complicated dynamics of rowmotion on $C_{n}\times [k]$, where $C_{n}$ is the claw poset with $n$ unrelated elements each covering $\widehat{0}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07984 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rowmotion on the chain of V's poset and whirling dynamics Plante, Matthew Roby, Tom Combinatorics Given a finite poset $P$, we study the _whirling_ action on vertex-labelings of $P$ with the elements $\{0,1,2,\dotsc ,k\}$. When such labelings are (weakly) order-reversing, we call them $k$-bounded $P$-partitions. We give a general equivariant bijection between $k$-bounded $P$-partitions and order ideals of the poset $P\times [k]$ which conveys whirling to the well-studied rowmotion operator. As an application, we derive periodicity and homomesy results for rowmotion acting on the chain of V's poset $V \times [k]$. We are able to generalize some of these results to the more complicated dynamics of rowmotion on $C_{n}\times [k]$, where $C_{n}$ is the claw poset with $n$ unrelated elements each covering $\widehat{0}$. |
| title | Rowmotion on the chain of V's poset and whirling dynamics |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.07984 |