# On formulas and some combinatorial properties of Schubert Polynomials

## TL;DR

This paper introduces two new formulas for Schubert polynomials using Gröbner-Shirshov bases, explores their combinatorial properties, and provides algorithms for calculating their structure constants, enhancing computational methods in algebraic combinatorics.

## Contribution

It presents novel formulas for Schubert polynomials based on Gröbner-Shirshov bases and develops algorithms for their structure constants, including one utilizing Monk's formula.

## Key findings

- Two formulas for Schubert polynomials involving only nonnegative monomials
- Proved combinatorial properties of Schubert polynomials
- Developed algorithms for calculating structure constants

## Abstract

By applying a Gr\"{o}bner-Shirshov basis of the symmetric group $S_{n}$, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk's formula.