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Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq 1} \big(i!^{s_i(\pi)} s_i(\pi)!\big)},\] where $s_i(\pi)$ denotes the number of parts $i$ in $\pi$.

Question: Is there a book that contains a proof of this?

I'm looking to cite it in a paper and would prefer not to include a proof. I attempted a search in Google books, but that didn't help too much.

A similar result is proved in "Combinatorics: topics, techniques, algorithms" by Peter Cameron (page 212), but has "permutation" instead of "set partition" and "cycle structure" instead of "integer partition".

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up vote 4 down vote accepted

These are the coefficients in the expansion of power-sum symmetric functions in terms of augmented monomial symmetric functions. I believe you will find a proof in:

Peter Doubilet. On the foundations of combinatorial theory. VII. Symmetric functions through the theory of distribution and occupancy. Studies in Appl. Math., 51:377–396, 1972.

See also MacMahon

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Thanks for that. From your answer, I was able to find it in: The theory of partitions George E. Andrews (Theorem 13.2). – Douglas S. Stones Oct 7 '11 at 21:14

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