树子网络的三叉表：与双墙表的一一对应
Trident Tableaux for Tree-Child Networks with One Reticulation Node: A Bijection with Two-Wall Tableaux
- Department of Pure and Applied Mathematics, Waseda University（早稻田大学理工创成学研究科）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文引入三叉表，构造其与双墙表的一一对应，证明基数公式，并用于精确计数含一个网状节点的树子网络，其比例渐近于1/8。
AI中文摘要：
受树子网络的单词编码启发，我们引入了三叉表，这是一种具有唯一三单元格列且满足特定后继条件的杨表。我们构造了三叉表（具有$n+1$列）与双墙表（具有$n$列）之间的一一对应，双墙表是一种具有两个指定列且不施加垂直顺序的两行填充。该对应匹配了两个类别的三部分分解，并表明每个类别的基数为$n(n+1)C_n/2$，其中$C_n$是第$n$个加泰罗尼亚数。特别地，它给出了OEIS A002457的移位形式的三叉表解释。作为结果，我们获得了包含一个三叉的具有一个网状节点的树子网络的精确计数，并表明随着叶子数趋于无穷，它们在所有具有一个网状节点的树子网络中的比例趋于$1/8$。
英文摘要：
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.
