Richardson tableaux and noncrossing partial matchings

2025/11/19 by Guo, Peter L.
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications

paper · doi:10.48550/arxiv.2511.15094

openalex publication_date 2025/11/19 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28

Abstract

Richardson tableaux are a remarkable subfamily of standard Young tableaux introduced by Karp and Precup in order to index the irreducible components of Springer fibers equal to Richardson varieties. We show that the set of insertion tableaux of noncrossing partial matchings on \1,2,,…, n\ by applying the Robinson--Schensted algorithm coincides with the set of Richardson tableaux of size n. This leads to a natural one-to-one correspondence between the set of Richardson tableaux of size n and the set of Motzkin paths with n steps, in response to a problem proposed by Karp and Precup. As consequences, we recover some known and establish new properties for Richardson tableaux. Especially, we relate the q-counting of Richardson tableaux to q-Catalan numbers.

Citations

• Positroids, knots, and q,t-Catalan numbers
• Total positivity in Springer fibres
• Electroid varieties and a compactification of the space of electrical networks
• On the singularity of some special components of Springer fibers
• Crossings and Nestings of Matchings and Partitions

Cited by

• The insertion encoding of restricted growth functions

Related

• Richardson tableaux and components of Springer fibers equal to Richardson varieties
• Matchings Avoiding Partial Patterns
• Richardson tableaux and Schubert positivity
• Schröder partitions, Schröder tableaux and weak poset patterns
• Flag varieties and interpretations of Young tableau algorithms