Combinatorics

. 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 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.