A lattice on decreasing trees : the metasylvester lattice

TL;DR

This paper introduces the metasylvester lattice, a new combinatorial structure on decreasing trees, which generalizes the $m$-Tamari lattice and provides new realizations and insights into its properties.

Contribution

It defines the metasylvester lattice as a sublattice of $m$-permutations, connecting decreasing trees with $m$-Tamari lattices and offering new combinatorial realizations.

Findings

The metasylvester lattice is a sublattice of $m$-permutations.

It provides new realizations of the $m$-Tamari lattice.

The structure reveals combinatorial properties linking decreasing trees and $m$-generalizations.

Abstract

We introduce a new combinatorial structure: the metasylvester lattice on decreasing trees. It appears in the context of the $m$-Tamari lattices and other related $m$-generalizations. The metasylvester congruence has been recently introduced by Novelli and Thibon. We show that it defines a sublattice of the $m$-permutations where elements can be represented by decreasing labelled trees: the metasylvester lattice. We study the combinatorial properties of this new structure. In particular, we give different realizations of the lattice. The $m$-Tamari lattice is by definition a sublattice of our newly defined metasylvester lattice. It leads us to a new realization of the $m$-Tamari lattice, using certain chains of the classical Tamari lattice.