Combinatorial aspects of the Lascoux–Schützenberger tree

Abstract

In 1982, Richard Stanley introduced the formal series Fσ(X) in order to enumerate reduced decompositions of a given permutation σ. Stanley (European J. Combin. 5(4) (1984) 359) not only showed Fσ(X) to be symmetric, but in certain cases, Fσ(X) was a Schur function. Stanley conjectured that for arbitrary σ,Fσ(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 Fσ(X).