The Skeleton of a Reduced Word and a Correspondence of Edelman and Greene

Abstract

Stanley conjectured that the number of maximal chains in the weak Bruhat order of S n , or equivalently the number of reduced decompositions of the reverse of the identity permutation w 0 = n, n − 1, n − 2, . . . , 2, 1, equals the number of standard Young tableaux of staircase shape s = {n − 1, n − 2, . . . , 1}. Originating from this conjecture remarkable connections between standard Young tableaux and reduced words have been discovered. Stanley proved his conjecture algebraically, later Edelman and Greene found a bijective proof. We provide an extension of the Edelman and Greene bijection to a larger class of words. This extension is similar to the extension of the Robinson-Schensted correspondence to two line arrays. Our proof is inspired by Viennot's planarized proof of the Robinson-Schensted correspondence. As it is the case with the classical correspondence the planarized proofs have their own beauty and simplicity.