Richardson tableaux and Motzkin paths
Abstract
Richardson tableaux were introduced by Karp and Precup in their study of irreducible components of Springer fibers that are Richardson varieties. Guo gave an explicit bijection between Richardson tableaux and Motzkin paths through noncrossing involutions and the Robinson--Schensted (RS) algorithm. In this paper, we study Richardson tableaux of a fixed shape from the viewpoint of Motzkin paths. We introduce a shape algorithm, independent of the RS algorithm, that directly determines the shape of the corresponding Richardson tableau from a Motzkin path. Based on this algorithm, we construct a local bijection which keeps track of the major statistic and reproves combinatorially Karp and Precup's q-enumeration formula for Richardson tableaux of a given shape. We also prove in two ways a conjecture of Guo on the comajor generating function for Richardson tableaux with a prescribed number of odd columns.
Related
- Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity
- 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
- Baxter Structures and the Odd-Even Enumeration of Red-Blue Motzkin Paths
- The Cuntz-Pimsner algebra of the simplest Motzkin subproduct system is 2-subhomogeneous
- Motzkin Numbers Count 2-Stack-Sortable Permutations Ending in Their Least Entry
BibTeX
@misc{eu2026richardson,
title = {{Richardson tableaux and Motzkin paths}},
author = {Sen-Peng Eu and Tung-Shan Fu and Yi-Hao Kao and Yi-Lin Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.21245},
eprint = {2608.21245},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21245},
}
RIS
TY - UNPB
TI - Richardson tableaux and Motzkin paths
AU - Sen-Peng Eu
AU - Tung-Shan Fu
AU - Yi-Hao Kao
AU - Yi-Lin Lee
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2608.21245
UR - https://arxiv.org/abs/2608.21245
AB - Richardson tableaux were introduced by Karp and Precup in their study of irreducible components of Springer fibers that are Richardson varieties. Guo gave an explicit bijection between Richardson tableaux and Motzkin paths through noncrossing involutions and the Robinson--Schensted (RS) algorithm. In this paper, we study Richardson tableaux of a fixed shape from the viewpoint of Motzkin paths. We introduce a shape algorithm, independent of the RS algorithm, that directly determines the shape of the corresponding Richardson tableau from a Motzkin path. Based on this algorithm, we construct a local bijection which keeps track of the major statistic and reproves combinatorially Karp and Precup's q-enumeration formula for Richardson tableaux of a given shape. We also prove in two ways a conjecture of Guo on the comajor generating function for Richardson tableaux with a prescribed number of odd columns.
ER -