Pebble trees

TL;DR

This paper introduces pebble trees, a new combinatorial structure, and demonstrates their connection to a convex polytope called pebble tree polytope, generalizing classical polytopes like permutahedra and associahedra.

Contribution

It establishes an isomorphism between the contraction poset of pebble trees and the face poset of the pebble tree polytope, providing convex realizations of certain associahedra.

Findings

Pebble trees form a new combinatorial class with a specific pebble distribution rule.

The contraction poset of pebble trees is isomorphic to the face poset of the pebble tree polytope.

Provides convex polytope realizations of all associahedra constructed by Poirier and Tradler.

Abstract

A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that the contraction poset on pebble trees is isomorphic to the face poset of a convex polytope called pebble tree polytope. Beside providing intriguing generalizations of the classical permutahedra and associahedra, our motivation is that the faces of the pebble tree polytopes provide realizations as convex polytopes of all assocoipahedra constructed by K. Poirier and T. Tradler only as polytopal complexes.