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

TL;DR

This paper introduces the $s$-weak order and $s$-Tamari lattices, generalizing classical permutation orders and Tamari lattices, and establishes their combinatorial and lattice-theoretic properties.

Contribution

It defines the $s$-weak order and $s$-Tamari lattices, proving they are semidistributive, congruence uniform, and related to known structures like $ u$-Tamari lattices.

Findings

$s$-weak order is a semidistributive, congruence uniform lattice.

$s$-Tamari lattices are sublattices and quotients of the $s$-weak order.

$s$-Tamari lattices are isomorphic to $ u$-Tamari lattices.

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.