Number of Elements with given Cycle Type
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Theorem
Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\lambda$ be an integer partition of $n$ such that $\lambda$ has $a_j$ parts of size $j$ for each $j$.
That is, such that there are $a_1$ instances of $1$s, $a_2$ instances of $2$s, $a_3$ instances of $3$s, and so on.
Let $C$ be the conjugacy class in $S_n$ comprising the elements whose cycle type is $\lambda$.
Then:
- $\size C = \dfrac {n!} {\ds \prod_j j^{a_j} a_j!}$
Proof
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Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 80 \beta$