Some Combinatorial Properties of Schubert Polynomials

Abstract

Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial (Formula presented.) in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set (Formula presented.) of tableaux with w, each tableau contributing a single term to (Formula presented.). This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that (Formula presented.) is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function (Formula presented.), where (Formula presented.) denotes a skew Schur function. © 1993, Kluwer Academic Publishers. All rights reserved.