Stasheff polytope as a sublattice of permutohedron

TL;DR

This paper demonstrates that the Stasheff polytope (associahedron) can be embedded as a sublattice within the permutohedron, revealing a structural relationship between Tamari and permutation lattices.

Contribution

It establishes a natural association between the vertices of the associahedron and permutohedron that preserves lattice operations, showing Tamari lattices as sublattices of permutation lattices.

Findings

Vertices of associahedron correspond to specific vertices of permutohedron.

Lattice operations are preserved under this correspondence.

Tamari lattices are sublattices of permutation lattices.

Abstract

An assosiahedron $\mathcal{K}^n$, known also as Stasheff polytope, is a multifaceted combinatorial object, which, in particular, can be realized as a convex hull of certain points in $\mathbf{R}^{n}$, forming $(n-1)$-dimensional polytope. A permutahedron $\mathcal{P}^n$ is a polytope of dimension $(n-1)$ in $\mathbf{R}^{n}$ with vertices forming various permutations of $n$-element set. There exist well-known orderings of vertices of $\mathcal{P}^n$ and $\mathcal{K}^n$ that make these objects into lattices: the first known as permutation lattices, and the latter as Tamari lattices. We establish that the vertices of $\mathcal{K}^n$ can be naturally associated with particular vertices of $\mathcal{P}^n$ in such a way that the corresponding lattice operations are preserved. In lattices terms, Tamari lattices are sublattices of permutation lattices. More generally, this defines the application of associative law as a special form of permutation.