Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

TL;DR

This paper explores the combinatorial and algebraic structures of painted tree polytopes, linking face posets to graded Hopf algebras and providing new insights into graph associahedra and related polytopes.

Contribution

It introduces a framework connecting painted trees, face posets, and Hopf algebra structures, including new algebraic structures on graph tubings.

Findings

Face posets of painted tree polytopes form graded Hopf algebras.

Identifies the polytopes as classical permutohedra and generalized permutohedra.

Provides algebraic structures on graph tubings and star graphs.

Abstract

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.