Proof of a Conjecture on Young Tableaux with Walls

TL;DR

This paper proves a conjecture on the enumeration of Young tableaux with walls, leading to the resolution of a significant problem in phylogenetics related to tree-child networks.

Contribution

It establishes a proof for a conjecture on Young tableaux with walls and verifies a phylogenetics conjecture involving tree-child networks.

Findings

Proof of a conjecture on Young tableaux with walls

Verification of a phylogenetics conjecture

Advancement in combinatorial enumeration methods

Abstract

Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this work, we prove a conjecture of Fuchs and Yu concerning the enumeration of two classes of three-row Young tableaux with walls. Combining with the work by Chang, Fuchs, Liu, Wallner, and Yu leads to the verification of a conjecture on tree-child networks proposed by Pons and Batle. This conjecture was regarded as a specific and challenging problem in the Phylogenetics community until it was finally resolved by the present work.