Weighted enumeration of Bruhat chains in the symmetric group

Abstract

We use the recently introduced padded Schubert polynomials to prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is ( n 2 ) ! {n \choose 2}! for both the code weights and the Chevalley weights, generalizing a result of Stembridge. We also define weights which give a one-parameter family of strong order analogues of Macdonald’s well-known reduced word identity for Schubert polynomials.