Maximal degree subposets of $ν$-Tamari lattices
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911271888093184 |
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| author | Dermenjian, Aram |
| author_facet | Dermenjian, Aram |
| contents | In this paper, we study two different subposets of the $ν$-Tamari lattice: one in which all elements have maximal in-degree and one in which all elements have maximal out-degree. The maximal in-degree and maximal out-degree of a $ν$-Dyck path turns out to be the size of the maximal staircase shape path that fits weakly above $ν$. For $m$-Dyck paths of height $n$, we further show that the maximal out-degree poset is poset isomorphic to the $ν$-Tamari lattice of $(m-1)$-Dyck paths of height $n$, and the maximal in-degree poset is poset isomorphic to the $(m-1)$-Dyck paths of height $n$ together with a greedy order. We show these two isomorphisms and give some properties on $ν$-Tamari lattices along the way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_11417 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Maximal degree subposets of $ν$-Tamari lattices Dermenjian, Aram Combinatorics 05E, 05A19, 06A07 In this paper, we study two different subposets of the $ν$-Tamari lattice: one in which all elements have maximal in-degree and one in which all elements have maximal out-degree. The maximal in-degree and maximal out-degree of a $ν$-Dyck path turns out to be the size of the maximal staircase shape path that fits weakly above $ν$. For $m$-Dyck paths of height $n$, we further show that the maximal out-degree poset is poset isomorphic to the $ν$-Tamari lattice of $(m-1)$-Dyck paths of height $n$, and the maximal in-degree poset is poset isomorphic to the $(m-1)$-Dyck paths of height $n$ together with a greedy order. We show these two isomorphisms and give some properties on $ν$-Tamari lattices along the way. |
| title | Maximal degree subposets of $ν$-Tamari lattices |
| topic | Combinatorics 05E, 05A19, 06A07 |
| url | https://arxiv.org/abs/2208.11417 |