Stasheff polytope as a sublattice of permutohedron

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.