Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Arkadii Slinko

TL;DR

The paper identifies a hidden Condorcet domain in Loday's realization of the associahedron by examining its intersection with the permutohedron. It proves that $\mathrm{Perm}_n \cap \mathrm{Asso}_n$ equals the nested-star domain of size $2^{n-1}$, forming a maximal symmetric never-middle Condorcet domain, with a base case at $n=3$ and an inductive argument using a root-weight $w(v)=(i+1)(n-i)$. This weight-based inductive analysis restricts which trees contribute to permutation vertices, establishing the exact count and structure of common vertices. The result connects polyhedral realizations to voting-theoretic domains and provides a combinatorial description of a large Condorcet domain embedded in Loday's associahedron.

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.

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

TL;DR

The paper identifies a hidden Condorcet domain in Loday's realization of the associahedron by examining its intersection with the permutohedron. It proves that equals the nested-star domain of size , forming a maximal symmetric never-middle Condorcet domain, with a base case at and an inductive argument using a root-weight . This weight-based inductive analysis restricts which trees contribute to permutation vertices, establishing the exact count and structure of common vertices. The result connects polyhedral realizations to voting-theoretic domains and provides a combinatorial description of a large Condorcet domain embedded in Loday's associahedron.

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.
Paper Structure (2 sections, 3 equations, 1 figure)

This paper contains 2 sections, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Permutations and binary trees.

Figures (1)

  • Figure 1: