Two-row alternating character cancellation
= Two-row alternating character cancellation
For even $n$, the <virtual character>
$$
F_n=\sum_{r=0}^{n/2}(-1)^r\chi^{(n-r,r)}
$$
vanishes on every permutation having an odd cycle. Under the <Frobenius characteristic map>, the <Jacobi–Trudi identity> identifies its characteristic with the degree-$n$ part of $H(t)H(-t)$, which contains only products of even-indexed power sums.