On the Forking Path Conjecture

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 . (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.