Let $C\_\mu$ be the size of the conjugacy class in $S\_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S\_n$.


Let $\omega\vdash m$ and let $\theta\vdash(n+m)$. I am interested in the sum 
$$ \frac{1}{n!}\sum\_{\mu\vdash n} C\_\mu\chi\_\lambda(\mu)\chi\_\theta(\mu\cup\omega),$$ where $\mu\cup\omega$ is a partition of $n+m$ containing the parts of $\mu$ and of $\omega$. Numerics suggest that for most pairs $(\lambda,\theta)$ this sum is zero. In the simplest case of $\omega=0$, for example, the only non-vanishing pair is $\theta=\lambda$ (and the result is 1).


To illustrate, these are the values of the sum when $\omega=(3)$, with $n=4$ and $n+m=7$ ($\lambda$ is labelling the rows and $\theta$ the columns, both in lexicographic order)
$$\left[ \begin {array}{ccccccccccccccc} 1&0&0&1&0&-1&0&0&1&0&0&0&0&0&0
\\ 0&1&0&0&0&0&0&-1&0&0&0&1&0&0&0
\\ 0&0&1&-1&0&0&0&0&0&0&-1&0&1&0&0
\\ 0&0&0&0&1&0&-1&0&0&0&0&0&0&1&0
\\ 0&0&0&0&0&0&0&0&1&-1&1&0&0&0&1\end {array}
 \right] $$


A possible solution would be to write $\chi\_\theta(\mu\cup\omega)=\sum\_{\rho\vdash n} a\_\rho(\omega) \chi\_\rho(\mu)$. Is there a known way to accomplish this decomposition? (I'm thinking something like the Murnaghan-Nakayama rule)



A sum over characters of the symmetric group
Since $\chi\_\theta(\mu\cup\omega)=\langle s\_\theta,p\_\mu p\_\omega\rangle$, your sum is given by
 $$ \frac{1}{n!}\sum\_{\mu\vdash n}
 C\_\mu\chi\_\lambda(\mu)\langle s\_\theta,p\_\mu p\_\omega\rangle =
 \left\langle s\_\theta,p\_\omega\cdot \frac{1}{n!}\sum\_{\mu\vdash
 n} C\_\mu \chi\_\lambda(\mu)p\_\mu\right\rangle $$
 $$ \qquad\qquad = \langle s\_\theta,p\_\omega s\_\lambda\rangle. $$
We can then expand $p\_\omega s\_\lambda$ in terms of Schur functions by
Theorem 7.17.1 (the basis for the Murnaghan-Nakayama rule) of
*Enumerative Combinatorics*, vol. 2. Note also that $\langle s\_\theta,p\_\omega s\_\lambda\rangle = \langle s\_{\theta/\lambda},p\_\omega\rangle =
\chi\_{\theta/\lambda}(\omega)$, a value of the skew character $\chi\_{\theta/\lambda}$.


