 Solution

For $F(t)={\prod }_i(1+x_{i}t)$ the coefficients are the elementary symmetric functions, $f_r=e_r(x)$. The specialization is evaluation in $x$ after the symmetric-function involution, because $\omega (h_r)=e_r$. HenceOne can also read this coefficient directly from the dual Cauchy identity, after exchanging its alphabets. For a general symmetric function, $u^F=(\omega u)(x)$; it is not simply $u(x)$.