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