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Is there a formula for $\#$cycle types in $S_n$?

Example: all cyle-types for $S_5$: \begin{align} &(5)\\ &(41)\\ &(32)\\ &(311)\\ &(221)\\ &(2111)\\ &(11111)\\ \end{align}

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up vote 3 down vote accepted

The number of cycle types of $S_n$ is the number of ways to "partition" $n$, and there is no nice formula for this. See this wikipedia page for more details:

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There is one cycle type in $S_n$ for each partition of $n$. Unfortunately, there is no nice, simple formula for the partition function.

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