Trident Tableaux for Tree-Child Networks with One Reticulation Node: A Bijection with Two-Wall Tableaux
Abstract
Motivated by a word encoding of tree-child networks, we introduce trident tableaux, which are Young tableaux with a unique three-cell column satisfying certain conditions on successors. We construct a bijection between trident tableaux with n+1 columns and two-wall tableaux with n columns, that is, two-row fillings with two designated columns in which vertical order is not imposed. The bijection matches the three-part decompositions of the two classes and shows that each class has cardinality n(n+1)Cn/2, where Cn is the n-th Catalan number. In particular, it gives a trident-tableau interpretation of a shifted form of OEIS A002457. As a consequence, we obtain an exact enumeration of tree-child networks with one reticulation node that contain a trident, and show that their proportion among all tree-child networks with one reticulation node tends to 1/8 as the number of leaves tends to infinity.
Related
- Type C transpositions and strong marked tableaux
- Characterization of k-highest weight and ĝ-dominant tableaux via $1−0$-slack recording tableaux in the quantum Littlewood-Richardson rule and lattice points in flagged hive polytopes
- The tableaux monoid: a rewriting-theoretic approach
- River ribbon tableaux and symmetric group characters
- Kashiwara--Nakashima tableaux, Gelfand--Tsetlin patterns, and quantum symmetric pairs
BibTeX
@misc{liu2026trident,
title = {{Trident Tableaux for Tree-Child Networks with One Reticulation Node: A Bijection with Two-Wall Tableaux}},
author = {Hexuan Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28892},
eprint = {2609.28892},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28892},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Trident Tableaux for Tree-Child Networks with One Reticulation Node: A Bijection with Two-Wall Tableaux
AU - Hexuan Liu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28892
UR - https://arxiv.org/abs/2609.28892
AB - Motivated by a word encoding of tree-child networks, we introduce trident tableaux, which are Young tableaux with a unique three-cell column satisfying certain conditions on successors. We construct a bijection between trident tableaux with n+1 columns and two-wall tableaux with n columns, that is, two-row fillings with two designated columns in which vertical order is not imposed. The bijection matches the three-part decompositions of the two classes and shows that each class has cardinality n(n+1)C_n/2, where C_n is the n-th Catalan number. In particular, it gives a trident-tableau interpretation of a shifted form of OEIS A002457. As a consequence, we obtain an exact enumeration of tree-child networks with one reticulation node that contain a trident, and show that their proportion among all tree-child networks with one reticulation node tends to 1/8 as the number of leaves tends to infinity.
ER -