Cyclic sieving phenomena for trees and tree-rooted maps
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arXiv
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| Natura: | Preprint |
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2025
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| author | Bousquet-Mélou, Mireille Krattenthaler, Christian |
| author_facet | Bousquet-Mélou, Mireille Krattenthaler, Christian |
| contents | We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees with a given degree distribution of the nodes. Moreover, we consider four different cyclic group actions: (1) the root is moved to the next corner along a tour of the tree; (2) only trees in which the root is at a leaf are considered, and the action moves the root to the next leaf; (3) only trees in which the root is at a non-leaf are considered, and the action moves the root to the next non-leaf corner; (4) only trees in which the root is at a node of degree $δ$ are considered, for a fixed $δ$, and the action moves the root to the next corner of this type. We prove a cyclic sieving phenomenon for each meaningful combination of these sets and actions. As a bonus, we also establish corresponding cyclic sieving phenomena for tree-rooted planar maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cyclic sieving phenomena for trees and tree-rooted maps Bousquet-Mélou, Mireille Krattenthaler, Christian Combinatorics 05A15, 05A10 We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees with a given degree distribution of the nodes. Moreover, we consider four different cyclic group actions: (1) the root is moved to the next corner along a tour of the tree; (2) only trees in which the root is at a leaf are considered, and the action moves the root to the next leaf; (3) only trees in which the root is at a non-leaf are considered, and the action moves the root to the next non-leaf corner; (4) only trees in which the root is at a node of degree $δ$ are considered, for a fixed $δ$, and the action moves the root to the next corner of this type. We prove a cyclic sieving phenomenon for each meaningful combination of these sets and actions. As a bonus, we also establish corresponding cyclic sieving phenomena for tree-rooted planar maps. |
| title | Cyclic sieving phenomena for trees and tree-rooted maps |
| topic | Combinatorics 05A15, 05A10 |
| url | https://arxiv.org/abs/2512.18656 |