A simple permutoassociahedron

TL;DR

This paper introduces a family of simple permutoassociahedra, combinatorial structures that are convex polytopes combining features of permutohedra and associahedra, useful for topological proofs.

Contribution

It constructs a new family of simple permutoassociahedra, extending previous work on these hybrid polytopes with potential applications in topology.

Findings

Constructed a family of simple permutoassociahedra

Provided a topological proof of Mac Lane's coherence

Extended the class of known convex polytopes

Abstract

In the early 1990s, a family of combinatorial CW-complexes named permutoassociahedra was introduced by Kapranov, and it was realized by Reiner and Ziegler as a family of convex polytopes. The polytopes in this family are "hybrids" of permutohedra and associahedra. Since permutohedra and associahedra are simple, it is natural to search for a family of simple permutoassociahedra, which is still adequate for a topological proof of Mac Lane's coherence. This paper presents such a family.