Richardson tableaux and noncrossing partial matchings

TL;DR

The paper proves that the insertion tableaux of noncrossing partial matchings on $[n]$ coincide with the Richardson tableaux of size $n$, yielding a bijection from NC(n) to RT(n). It constructs this correspondence by applying the Robinson–Schensted algorithm (RS) to noncrossing involutions and analyzes prime versus composite cases via concatenation of tableaux. Consequences include a constructive Motzkin-path interpretation and a link to $q$-Catalan numbers, with an explicit $q$-enumeration for even Richardson tableaux. These results provide new combinatorial tools for studying Springer fibers and connect geometry to explicit tableau–Motzkin correspondences.

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 $\{1,2,,\ldots, n\}$ 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.

Figures (8)

• Figure 1.1: Two partial matchings on $[8]$.
• Figure 1.2: Diagrams of $w$ and $w_0ww_0$ for $w=52431687$.
• Figure 2.3: Two semistandard Young tableaux of size $7$.
• Figure 2.4: An illustration of the RSK algorithm.
• Figure 3.5: A prime noncrossing partial matching.

Theorems & Definitions (23)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3: KP
• proof
• Corollary 1.4: KP
• proof
• Corollary 1.5
• proof
• Corollary 1.6
• proof