On the Forking Path Conjecture

TL;DR

This paper proves the Forking Path Conjecture for most elements in S_4, identifies a counterexample, and proposes a refined conjecture for the longest element in any symmetric group.

Contribution

It advances understanding of the Forking Path Conjecture by proving it for all but one element in S_4 and suggesting a refined conjecture for larger symmetric groups.

Findings

Proved the conjecture for all but one element in S_4

Identified a counterexample involving two specific paths

Proposed a refined conjecture for the longest element in S_n

Abstract

We prove the Forking Path Conjecture for all but one element in the symmetric group $S_4$. Two specific paths in the rex graph of that element give a counterexample for the conjecture. We propose a refined conjecture for the longest element of any $S_n$.