Determinantal Formulas for SEM Expansions of Schubert Polynomials

We show that for any permutation w avoids a certain set of 13 patterns length 5 and 6, the Schubert polynomial \({\mathfrak {S}}_w\) can be expressed as determinant matrix elementary symmetric polynomials in manner similar to Jacobi–Trudi identity. For such w, this determinantal formula is equivalent (signed) subtraction-free expansion \(\mathfrak S_w\) basis standard monomials.