Richardson tableaux are exactly the tableaux whose Springer fiber component is a Richardson variety, and they are counted by Motzkin numbers.
Nil-Hecke rings and the Schubert calculus
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abstract
The purpose of the present notes is to give a self-contained exposition on the use of the techniques of Nil-Hecke algebras in the localization approach to the equivariant Schubert calculus for cohomology of flag varieties. We also demonstrate how these techniques can be applied to non-crystallographic root systems as well as to study (connective) $K$-theory of flag varieties.
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Richardson tableaux and components of Springer fibers equal to Richardson varieties
Richardson tableaux are exactly the tableaux whose Springer fiber component is a Richardson variety, and they are counted by Motzkin numbers.