Permutrees

TL;DR

Permutrees are a new combinatorial structure that unifies various well-known objects like permutations, trees, and sequences, with applications in lattice theory, polytope geometry, and algebraic Hopf structures.

Contribution

The paper introduces permutrees as a comprehensive model linking multiple combinatorial, geometric, and algebraic structures, unifying existing theories and structures.

Findings

Unified rotation lattices for permutations, trees, and sequences

Polytope descriptions including permutahedron and associahedra

Hopf algebra encompassing various combinatorial objects

Abstract

We introduce permutrees, a unified model for permutations, binary trees, Cambrian trees and binary sequences. On the combinatorial side, we study the rotation lattices on permutrees and their lattice homomorphisms, unifying the weak order, Tamari, Cambrian and boolean lattices and the classical maps between them. On the geometric side, we provide both the vertex and facet descriptions of a polytope realizing the rotation lattice, specializing to the permutahedron, the associahedra, and certain graphical zonotopes. On the algebraic side, we construct a Hopf algebra on permutrees containing the known Hopf algebraic structures on permutations, binary trees, Cambrian trees, and binary sequences.