I'm looking for a text I could cite that explicitly states the following result: for $\chi^\lambda$ the irreducible character of the symmetric group indexed by the partition $\lambda$, and for $\sigma \in \mathfrak{S}_n$ and $\lambda \vdash n$,
$\chi^{\lambda'}(\sigma) = (-1)^{n-\ell(\sigma)} \chi^\lambda(\sigma),$
where $\lambda'$ is the dual partition, and $\ell(\sigma)$ is the length of the cycle type partition associated to $\sigma$.
It is not a hard result to prove (a simple combinatorial method, for instance, is to invoke the $\omega$ involution from Chapter 7 of Stanley's Enumerative Combinatorics). What I'm hoping for is a text that states this result directly (even if as an exercise) that could be cited without introducing background material and breaking the flow of a paper whose main topic is somewhat different from combinatorial representation theory.
