A hidden Condorcet domain in Loday's realisation of the associahedron

TL;DR

This paper demonstrates that Loday's associahedron shares a large set of points with the permutohedron, forming a maximal symmetric Condorcet domain within the context of the Tamari lattice.

Contribution

It reveals a hidden Condorcet domain in Loday's associahedron, connecting it with the permutohedron and expanding understanding of their geometric and combinatorial relationship.

Findings

Identifies 2^{n-1} common points between associahedron and permutohedron.

These points form a maximal symmetric Condorcet domain.

Provides new insights into the structure of Loday's realization of the Tamari lattice.

Abstract

We prove that Loday's polytopal realisation of the nth Tamari lattice T_n, called associahedron, has 2^{n-1} common points with the permutohedron, which form a maximal never-middle (symmetric) Condorcet domain.