Combinatorial aspects of the Lascoux–Schützenberger tree

Abstract

In 1982, Richard Stanley introduced the formal series F_σ(X) in order to enumerate reduced decompositions of a given permutation σ. Stanley (European J. Combin. 5(4) (1984) 359) not only showed F_σ(X) to be symmetric, but in certain cases, F_σ(X) was a Schur function. Stanley conjectured that for arbitrary $\text{σ,}\text{F}{}_{\mathrm{\sigma }}\text{(X)}$ was always Schur positive. Edelman and Greene subsequently proved this fact (Combinatories and Algebra (Boulder, CO, 1983), Amer. Math. Soc., Providence RI, 1984, pp. 155–162; Adv. in Math. 63(1) (1987) 42). Using the techniques of Lascoux and Schützenberger (Lett. Math. Phys. 10(2–3) (1985) 111) for computing Littlewood–Richardson coefficients, we will exhibit a new bijective proof of the Schur positivity of F_σ(X).