# Conjugacy classes of symmetric groups

I’m preparing a talk about representations of the symmetric group so I’ll write up some of it on this blog.

Recall that every permutation can be uniquely decomposed (modulo the order of the cycles) in a product of disjoint cycles. i.e.

for some , where . Here for example the cycle represents the permutation of the set that takes 1 to 2, 2 to 3, 3 to 4 and 4 to 1. If is some cycle of a permutation in and , then

.

Indeed, let . If for some , then . If not, i.e. if , then .

So for some with its decomposition in cycles as above, we have, for ,

.

This shows that the number of cycles in the cycle decomposition of is preserved under conjugation. Conversely, if and have the same number of cycles for all in their cycle decomposition, there is an obvious permutation such that . What we get from this is the fact that the “cycle type” completely determines the conjugacy classes in .

For a positive integer , a *partition of * is an ordered set of integers such that . The notation indicates that is a partition of .

For example, is a partition of . One way to compactify the notation would be to write this partition as .

There is an obvious bijective correspondence between cycle types of permutations in and partitions of : to the permutation is associated the partition . The conjugacy classes of are thus indexed by the partitions of .

The function that associates to a positive integer the number of partitions of has been extensively studied. There is no known closed form for this but according to Wikipedia, Hardy & Ramanujan obtained in 1918 an asymptotic formula for as :

.

By an easy counting argument, we find that for a given partition of , the number of elements of in the associated conjugacy class is

.

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