# Proof of an entropy conjecture of Leighton and Moitra

## TL;DR

This paper proves a conjecture by Leighton and Moitra, establishing an entropy bound for probability distributions on permutations that favor certain tournament arcs, with improvements for transitive tournaments.

## Contribution

The paper confirms the conjecture, providing a general proof and a sharper entropy bound for transitive tournaments, advancing understanding of permutation distributions in combinatorics.

## Key findings

- Entropy of distributions exceeds a threshold if they favor tournament arcs.
- For transitive tournaments, a shorter proof with a better entropy bound is provided.
- The entropy bound depends on a fixed positive parameter related to arc bias.

## Abstract

We prove the following conjecture of Leighton and Moitra. Let $T$ be a tournament on $[n]$ and $S_n$ the set of permutations of $[n]$. For an arc $uv$ of $T$, let $A_{uv}=\{\sigma \in S_n \, : \, \sigma(u)<\sigma(v) \}$. $\textbf{Theorem.}$ For a fixed $\varepsilon>0$, if $\mathbb{P}$ is a probability distribution on $S_n$ such that $\mathbb{P}(A_{uv})>1/2+\varepsilon$ for every arc $uv$ of $T$, then the binary entropy of $\mathbb{P}$ is at most $(1-\vartheta_{\varepsilon})\log_2 n!$ for some (fixed) positive $\vartheta_\varepsilon$. When $T$ is transitive the theorem is due to Leighton and Moitra; for this case we give a short proof with a better $\vartheta_\varepsilon$.