Motzkin 数计数以最小项结尾的 2-栈可排序排列
Motzkin Numbers Count 2-Stack-Sortable Permutations Ending in Their Least Entry
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中文总结 AI 辅助
本文证明了 Zhang 的猜想：以最小项结尾的 2-栈可排序排列数等于 Motzkin 数，并通过与标准杨表的双射给出证明。
中文摘要 AI 辅助
我们证明了 Zhang 的一个猜想（arXiv:2604.10779，猜想 6.1）：对于 $n \geq 0$，集合 $\{0,1,\dots,n\}$ 的以 $0$ 结尾的 $2$-栈可排序排列的数量是第 $n$ 个 Motzkin 数。根据 Zhang 的结果，以最小元素结尾的 $2$-栈可排序排列与宽度至多为 $2$ 的标准组合杨表之间存在双射。然后我们用双射方法证明，这些排列与宽度至多为 $3$ 的标准杨表数量相等，而已知后者由 Motzkin 数计数。
英文摘要
We prove the following conjecture of Zhang (arXiv:2604.10779, Conjecture 6.1): for $n \geq 0$, the number of $2$-stack-sortable permutations of $\{0,1,\dots,n\}$ ending in $0$ is the $n$th Motzkin number. By Zhang's result, there is a bijection between $2$-stack-sortable permutations ending in their least element and standard composition tableaux of width at most $2$. We then show bijectively that there are an equal number of these and standard Young tableaux of width at most $3$, which are known to be counted by the Motzkin numbers.
