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arXiv · 2212.11556

The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure

Cesar Ceballos·Viviane Pons
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Abstract

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 $\nu$-Tamari lattices of Pr\'eville-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.

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Cesar Ceballos, Viviane Pons. 2022-12-22. The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure. https://doi.org/10.1137/23m1605818

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