Associahedra, cyclohedra and inversion of power series

TL;DR

This paper introduces a new Hopf monoid structure involving cycles and paths, providing explicit formulas for antipodes and character inversion using combinatorial and geometric tools like tubings, noncrossing partitions, and graph associahedra.

Contribution

It defines a novel Hopf monoid of cycles and paths, extending the Faà di Bruno monoid, and connects algebraic inversion formulas to geometric structures such as associahedra and cyclohedra.

Findings

Provides cancellation-free antipode formulas

Describes the group of characters via pairs of power series

Relates character inversion to faces of associahedra and cyclohedra

Abstract

We introduce the Hopf monoid of sets of cycles and paths, which contains the Fa\`a di Bruno Hopf monoid as a submonoid. We give cancellation-free and grouping-free formulas for its antipode, one in terms of tubings and one in terms of \emph{pointed} noncrossing partitions. We provide an explicit description of the group of characters of this Hopf monoid in terms of pairs of power series. Using graph associahedra, we relate paths and cycles to associahedra and cyclohedra, respectively. We give formulas for inversion in the group of characters in terms of the faces of these polytopes.