An extension of Tamari lattices

TL;DR

This paper generalizes Tamari lattices by constructing posets Tam(v) for paths v, showing their duality and lattice structure, and connecting them to binary trees and symmetric group combinatorics.

Contribution

It introduces a new family of Tamari-like lattices Tam(v), proves their duality and lattice properties, and links them to binary tree rotations and symmetric group combinatorics.

Findings

Tam(v) is a lattice for any path v.

Tam(v) is dual to Tam(\overleftarrow{v}).

The classical Tamari lattice partitions Tam(v) lattices.

Abstract

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $v$, from which the duality follows easily. This also shows that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all the paths on the square grid that consist of $n-1$ unit steps. We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.