In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, the question was about this total sum: $$\frak{s}_n:=\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}=?$$
This time around, I wish to ask:
QUESTION 1. If $\mu$ and $\lambda$ run through partitions of distinct parts of $n$, then what is the value of the sum $$\frak{t}_n:=\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}=? \tag1$$
QUESTION 2. What is a conceptual or representation-theoretic interpretation of (1)?
