Some Combinatorial Properties of Schubert Polynomials

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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 𝔖w 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 ℳw of tableaux with w, each tableau contributing a single term to 𝔖w. 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 𝔖w is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function sλ/μ(1, q, q2, …), where sλ/μ denotes a skew Schur function.

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