Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of the Murnaghan-Nakayama rule that:
$$\chi^{\lambda}_{(\mu,1^{n-|\mu|})}=\sum_{\alpha\vdash |\mu|}\chi_{\mu}^{\alpha} f(\lambda\backslash \alpha),$$
where $\chi_\mu^\lambda$ is the irreducible character. This is invertible, giving:
$$f(\lambda\backslash \alpha) =\sum_{\mu\vdash k}\frac{\chi_\mu^\alpha}{z_\mu}\chi_{\mu 1^{n-|\mu|}},$$
where $z_\mu$ is the central character.
I'm trying to understand the nature of $f_m:=f(\lambda\backslash 1^{r-m})$ for $m=1,2,\cdots,r$. Obviously these can be computed by the Aitken determinent formula but I'm much more interested in them from a representation theory point of view. Specifically I'd like to consider restrictions of the above sum to just $f_m$. So for example, is there an interpretation for:
$$(?)=\sum_{m=1}^{\lambda_1}\chi_{\mu}^{\alpha} f_m?$$
I'm not sure if it's wise to just fix $\mu$ here. Have such things been considered before?
