I've been reading up on combinatorics in my spare time recently. Suppose $\lambda$ is a partition of $n$, and $f_k(\lambda)$ the number of times that $k$ appears as a part of $\lambda$. Also, let $g_k(\lambda)$ be the number of distinct parts of $\lambda$ which occur at least $k$ times in $\lambda$. Why is it that $\sum f_k(\lambda)=\sum g_k(\lambda)$ for fixed $k$, if we let the sums run over all partitions $\lambda$ of a fixed $n$?
I think the general idea is that for any partition $\lambda$ of $n$, and any part $p$ occurring at least $k$ times, one needs to associate some partition $\delta$ of $n$ so that the total number of times given $\delta$ occurs is the same as the number of parts of $\delta$ which are equal to $k$. Is there a way to flesh this out? Thank you.