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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.

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    I don't understand. Did you mean to write "$f_k(\lambda)$ is the number of times a *block of size* $k$ appears in $\lambda$?" (I don't see how $k$ itself can appear more than once in a given partition.)2012-02-16
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    @williamdemeo Yes, I mean it in that sense. For example, taking $n=13$ and $k=3$, then for the partition $\lambda=(1,1,2,3,3,3)$, I would say $f_3(\lambda)=3$.2012-02-16
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    Oh, okay. I was thinking of partitions of an $n$-element set; e.g. for $n=3$, $\lambda = |0|1,2|$ is a partition.2012-02-16
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    Your second paragraph doesn't read well. The number of distinct combinations $(\lambda,p)$ should equal the total number of occurrences of $k$, so "so that the total number of times given $\delta$ occurs is the same as the number of parts of $\delta$ which are equal to $k$" should be just "and an occurrence of a part $k$ in $\delta$".2012-02-16

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