= Solution
Apply the <Frobenius characteristic map>. For even $n$, the <Jacobi–Trudi identity> and cancellation of consecutive terms give
$$
\operatorname{ch}(F)
=\sum_{r=0}^{n}(-1)^r h_{n-r}h_r
=[t^n]H(t)H(-t).
$$
The generating function for the <complete homogeneous symmetric polynomials> now gives
$$
H(t)H(-t)
=\exp\left(\sum_{j\geq1}\frac{p_jt^j}{j}\right)
\exp\left(\sum_{j\geq1}\frac{(-1)^jp_jt^j}{j}\right)
=\exp\left(\sum_{j\geq1}\frac{p_{2j}t^{2j}}{j}\right).
$$
This expansion contains only products $p_\mu$ for which every part of $\mu$ is even. The coefficient of $p_\mu$ in $\operatorname{ch}(F)$ is $F(\mu)/z_\mu$, so $F(\mu)=0$ whenever the cycle type $\mu$ has an odd part. Equivalently, $F(g)=0$ whenever $g$ contains an odd cycle. This is the <Two-row alternating character cancellation>.
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