Let $\chi_{d,p;f}$ be the following symmetric polynomial,
$$\chi_{d,p;f}(x)=\prod_{l=1}^d\sum_{k=0}^p x_l^{f_k},$$
where $f=\lbrace f_0,\ldots,f_p\rbrace$ is a set of integers. I need to identify for which $f$ values $\chi_{d,p;f}$ is Schur-positive, i.e.:
$$\chi_{d,p;f}(x)=\sum_\rho a_\rho S_\rho(x_1,\ldots,x_d),$$ for $a_\rho$ non-negative integer. I've tried using the orthogonality of the Schur functions under the Haar measure of $U(n)$ to compute the coefficients explicitly $a_\rho=a_\rho(f)$ but the resulting expression is a complicated determinant of a Töplitz$\pm$Hankel matrix. There is any other way to find the Schur-positive cases for this polynomial? or at least some non-trivial $f$ examples?