The \(s\)-weak order and \(s\)-permutahedra. I: Combinatorics and lattice structure (Q6644116)

The Tamari lattice is a lattice quotient of the weak order that can be realized as a generalized permutahedron, and it was proven in [\textit{V. Pilaud} and \textit{F. Santos}, Bull. Lond. Math. Soc. 51, No. 3, 406--420 (2019; Zbl 1420.52015)] that every lattice quotient of the weak order satisfies this property.\N\NIn the paper under review (which is the first in a series) the authors introduce the notions of \(s\)-weak order and \(s\)-permutahedra, certain discrete objects that are indexed by a sequence of nonnegative integers \(s\). They concentrate 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, they show that the \(s\)-weak order is a semidistributive and congruence uniform lattice. Restricting the \(s\)-weak order to certain trees gives rise to the \(s\)-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. The authors 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 are isomorphic to the \(\nu\)-Tamari lattices of \textit{L. F. Préville-Ratelle} and \textit{X. Viennot} ([Trans. Am. Math. Soc. 369, No. 7, 5219--5239 (2017; Zbl 1433.05323)]).