The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure
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| Format: | Preprint |
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2022
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| author | Ceballos, Cesar Pons, Viviane |
| author_facet | Ceballos, Cesar Pons, Viviane |
| contents | This is the first contribution of a sequence of papers introducing the notions of $s$-weak order and $s$-permutahedra, certain discrete objects that are indexed by a sequence of non-negative integers $s$. In this first paper, we concentrate purely on the combinatorics and lattice structure of the $s$-weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the $s$-weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations.
Restricting the $s$-weak order to certain trees gives rise to the $s$-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the $s$-Tamari lattice can be obtained as a quotient lattice of the $s$-weak order when $s$ has no zeros, and show that the $s$-Tamari lattices (for arbitrary $s$) are isomorphic to the $ν$-Tamari lattices of Préville-Ratelle and Viennot.
The underlying geometric structure of the $s$-weak order will be studied in a sequel of this paper, where we introduce the notion of $s$-permutahedra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_11556 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure Ceballos, Cesar Pons, Viviane Combinatorics Primary 20F55, 06B05 and 06B10, Secondary 52B05 G.2.1 This is the first contribution of a sequence of papers introducing the notions of $s$-weak order and $s$-permutahedra, certain discrete objects that are indexed by a sequence of non-negative integers $s$. In this first paper, we concentrate purely on the combinatorics and lattice structure of the $s$-weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the $s$-weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations. Restricting the $s$-weak order to certain trees gives rise to the $s$-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the $s$-Tamari lattice can be obtained as a quotient lattice of the $s$-weak order when $s$ has no zeros, and show that the $s$-Tamari lattices (for arbitrary $s$) are isomorphic to the $ν$-Tamari lattices of Préville-Ratelle and Viennot. The underlying geometric structure of the $s$-weak order will be studied in a sequel of this paper, where we introduce the notion of $s$-permutahedra. |
| title | The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure |
| topic | Combinatorics Primary 20F55, 06B05 and 06B10, Secondary 52B05 G.2.1 |
| url | https://arxiv.org/abs/2212.11556 |