Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proof of a Conjecture on Young Tableaux with Walls

Zhicong Lin, Feihu Liu, Jiahang Liu, Jing Liu, Guoce Xin

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.

Proof of a Conjecture on Young Tableaux with Walls

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.
Paper Structure (18 sections, 35 theorems, 143 equations, 11 figures, 2 tables)

This paper contains 18 sections, 35 theorems, 143 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['MainConject']}
  3. A semi-closed-form expression for $a_{n,k}$
  4. An extension of $b_{n,k}$ and the introduction of its variant $\omega_{n,m,k}$
  5. Two key lemmas
  6. Finishing the proof of Theorem \ref{['MainConject']}
  7. Enumeration of tree-child networks
  8. Generating functions for $b_{n,k}$ and $b_{n,m,k}$
  9. An explicit formula for $D_k(t)$
  10. Functional equations for $B_k(x,t)$ via the kernel method
  11. Recurrence for $b_{n,k}$ via linear extensions of some posets
  12. Concluding remarks, open problems
  13. Appendix: Formulae for $a_{n,k}$ and $b_{n,k}$ for some fixed small $k$
  14. Appendix: Linear extensions of some posets
  15. $P$-partitions and linear extensions
  16. ...and 3 more sections

Key Result

Theorem 1.1

For $0\leq k\leq n$, we have

Figures (11)

  • Figure 1: A Young diagram with walls and a Young tableau with walls.
  • Figure 2: A Young diagram of $(7,7,4)$ with walls at the bottom.
  • Figure 3: A case in a family of deformed Young diagrams with walls for $n=7$ and $k=4$.
  • Figure 4: Two phylogenetic networks: only the right one is tree-child.
  • Figure 5: A case in a family of deformed Young diagrams with walls for $n=9$, $m=7$, and $k=4$.
  • ...and 6 more figures

Theorems & Definitions (61)

  • Theorem 1.1: Fuchs and Yu's enumerative conjecture
  • Conjecture 1.2: Pons and Batle PB
  • proof : Proof of Conjecture \ref{['conj:PB']}
  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.3
  • proof : Proof of Theorem \ref{['Semi-closed-formula-Ank']}
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 51 more