# Padded Schubert polynomials and weighted enumeration of Bruhat chains

## TL;DR

This paper generalizes the enumeration of maximal chains in the strong Bruhat order on symmetric groups, introducing weights that connect to Macdonald's identities and Schubert polynomials, offering new combinatorial insights.

## Contribution

It provides a unified framework for weighted enumeration of Bruhat chains, extending known formulas and introducing a one-parameter family of weights related to Schubert polynomials.

## Key findings

- Weighted counts of Bruhat chains generalize factorial formulas
- Introduction of weights leading to Macdonald's identities
- New combinatorial interpretations for Schubert polynomials

## Abstract

We prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is ${n \choose 2}!$ for both the code weights and the Chevalley weights. We also define weights which give a one-parameter family of strong order analogues of Macdonald's reduced word identity for Schubert polynomials.