Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity
Abstract
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,1n−2k).
Related
- Richardson tableaux and Motzkin paths
- Motzkin Numbers Count 2-Stack-Sortable Permutations Ending in Their Least Entry
- Entry and position statistics in alternating permutations and Young tableaux
- Inversion-descent enumerators of underline321-avoiding permutations
- Avoiding patterns with three distinct letters in Canon permutations
BibTeX
@misc{amaral2026motzkin,
title = {{Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity}},
author = {Ryan J. Amaral and Juan B. Gil and Michael D. Weiner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27096},
eprint = {2609.27096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27096},
}
RIS
TY - UNPB
TI - Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity
AU - Ryan J. Amaral
AU - Juan B. Gil
AU - Michael D. Weiner
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.27096
UR - https://arxiv.org/abs/2609.27096
AB - 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).
ER -