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Enumerative Combinatorics by Sergey Fomin, Richard P. Stanley

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Chapter 7: Appendix 1

Knuth Equivalence, Jeu de Taquin, and theLittlewood-Richardson Rule

(by Sergey Fomin)

This Appendix is devoted to the study of several combinatorial constructions involving standard Young tableaux (SYTs) that lead to the proof of the Littlewood-Richardson rule, a combinatorial rule describing the coefficients in the Schur function expansion of an arbitrary skew Schur function (or in a product of two ordinary Schur functions).

Most of what follows can be straightforwardly generalized to semistandard Young tableaux (SSYTs). We do not do it here, in order to simplify the presentation.

A1.1 Knuth Equivalence and Greene’s Theorem

The RSK algorithm (P, Q) associates to a permutation ∊ a pair of SYTs: the insertion tableau ...

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