Motzkin 路径、321-避免排列以及行奇偶性相同的标准杨表
Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity
- Penn State Altoona（宾夕法尼亚州立大学阿尔图纳分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文引入一类由 Motzkin 数计数的 321-避免排列，建立其与 Motzkin 路径及三行标准杨表的双射，使 Riordan 路径对应 Riordan 排列，并连接至更多 Riordan 数计数族。
AI中文摘要：
长度为 $n$ 的 Motzkin 路径和具有 $n$ 个格子且至多三行的标准杨表（SYT）都由 Motzkin 数计数，并且它们之间已知许多双射。Riordan 数计数 Riordan 路径（在 $x$-轴上没有水平步的 Motzkin 路径）的子族以及三行长度具有相同奇偶性的杨表的子族，但已知的双射都没有限制到这些子族。我们引入了 $321$-避免排列的集合，其中每个从左到右的最大值要么是一个下降，要么是一个不动点。这个族由 Motzkin 数计数，其无不动点元素是 Callan 的“Riordan 排列”。我们给出了一个从 Motzkin 路径到这些排列的双射，在该双射下 Riordan 路径对应于 Riordan 排列。然后我们给出了一个从这些排列到高度至多三的 SYT 的双射，该双射通过 Robinson–Schensted 插入后接奇偶性修正获得，在该双射下 Riordan 排列对应于行奇偶性相同的杨表，并且从左到右最大值的数量成为一个简单的杨表统计量。通过 Dyck 路径，我们将这些对象连接到由 Riordan 数计数的更多族，包括亏格为零的错排和形状为 $(k,k,1^{n-2k})$ 的 SYT。
英文摘要：
Motzkin paths of length $n$ and standard Young tableaux (SYT) with $n$ cells and at most three rows are both counted by the Motzkin numbers, and many bijections between them are known. The Riordan numbers count the subfamilies of Riordan paths (Motzkin paths with no horizontal step on the $x$-axis) and of tableaux whose three row lengths have the same parity, but none of the known bijections restricts to these subfamilies. We introduce the set of $321$-avoiding permutations in which every left-to-right maximum is either a descent or a fixed point. This family is counted by the Motzkin numbers, and its fixed-point-free elements are the ``Riordan permutations'' of Callan. We give a bijection from Motzkin paths to these permutations under which Riordan paths correspond to Riordan permutations. We then give a bijection from these permutations to SYT of height at most three, obtained from Robinson--Schensted insertion followed by a parity correction, under which Riordan permutations correspond to tableaux with rows of equal parity and the number of left-to-right maxima becomes a simple tableau statistic. Via Dyck paths, we connect these objects to further families counted by the Riordan numbers, including derangements of genus zero and SYT of shape $(k,k,1^{n-2k})$.
