Enumeration of labeled trees and Dyck tilings
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913465105383424 |
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| author | Shigechi, Keiichi |
| author_facet | Shigechi, Keiichi |
| contents | We study a partially ordered set of planar labeled rooted trees by use of combinatorial objects called Dyck tilings. A generating function of the poset is factorized when the minimum element of the poset is $312$-avoiding and satisfies some extra condition. We define a cover relation on rational Dyck tilings by that of labeled trees, and show that increasing and decreasing labelings are dual to each other. We consider two decompositions of a rational $(a,b)$-Dyck tiling: one is into $ab$ Dyck tilings and the other is into $a$ $(1,b)$-Dyck tilings. In the first case, we show that the weight of the $(a,b)$-Dyck tiling is the sum of the weights of $ab$ Dyck tilings. In the second case, we introduce a cover relation on $(1,b)$-Dyck tilings and obtain a poset of $(a,b)$-Dyck tilings by this decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05483 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enumeration of labeled trees and Dyck tilings Shigechi, Keiichi Combinatorics We study a partially ordered set of planar labeled rooted trees by use of combinatorial objects called Dyck tilings. A generating function of the poset is factorized when the minimum element of the poset is $312$-avoiding and satisfies some extra condition. We define a cover relation on rational Dyck tilings by that of labeled trees, and show that increasing and decreasing labelings are dual to each other. We consider two decompositions of a rational $(a,b)$-Dyck tiling: one is into $ab$ Dyck tilings and the other is into $a$ $(1,b)$-Dyck tilings. In the first case, we show that the weight of the $(a,b)$-Dyck tiling is the sum of the weights of $ab$ Dyck tilings. In the second case, we introduce a cover relation on $(1,b)$-Dyck tilings and obtain a poset of $(a,b)$-Dyck tilings by this decomposition. |
| title | Enumeration of labeled trees and Dyck tilings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.05483 |