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

Stanley conjectured that the number of maximal chains in the weak Bruhat order of Sn, or equivalently the number of reduced decompositions of the reverse of the identity permutation w0 = 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.