The Syntax Of Polytopal Projections: From Permutohedra To Associahedra
Combinatorics
2026-05-26 v1
Abstract
Tonks' projection from the permutohedron to the associahedron and the Loday--Ronco map both send permutations to planar binary trees. We give a syntactic account of these maps in the equational calculus of the free non-symmetric, non-unital operad on one binary generator. The vertex restriction of Tonks' projection is obtained by evaluating the head-insertion encoding on the reversed permutation, while the Loday--Ronco map is obtained by evaluating the decreasing encoding. We also give a local operadic proof that Tonks' vertex map is order-preserving from the weak Bruhat order to the Tamari order.
Cite
@article{arxiv.2605.25229,
title = {The Syntax Of Polytopal Projections: From Permutohedra To Associahedra},
author = {Miloš Adžić and Filip D. Jevtić},
journal= {arXiv preprint arXiv:2605.25229},
year = {2026}
}