I have asked this in MSE 8 days ago, even offered a bounty, and got nothing, so will try here.

I would like to understand the value of the skew characters of the symmetric group, $\chi_{\lambda/\mu}$ in the particular case when both $\lambda$ and $\mu$ are hooks, i.e. $\lambda=(a,1^{n-a})$ and $\mu=(b,1^{m-b})$ with $n>m$ and $a>b$.

As far as I can see, there would be two approaches to their calculation, but I'm a beginner and I cant work any of them through. I have looked around, but to no avail.

1) First, the Murnaghan–Nakayama rule says $$\chi_{\lambda/\mu}(\nu)=\sum_T (-1)^{{\rm ht}(T)},$$ where the sum is taken "over all border-strip tableaux of shape $\lambda/\mu$ and type $\nu$", according to Wikipedia. I do not really understand these tableaux. I mean, I think $\lambda/\mu$ is never a border strip if both are hooks; can I still sum over border strips of format $\lambda/\mu$?

2) I could also write $$\chi_{\lambda/\mu}(\nu)=\sum_\rho c^\lambda_{\mu\rho}\chi_\rho(\nu),$$ where $c^\lambda_{\mu\rho}$ are the Littlewood-Richardson coefficients. I looked around to see if they are known when $\lambda$ and $\mu$ are both hooks, but found nothing.