Richard Taylor ( once Ph.D. student of Andrew Wiles, the Fermat Theorem guy ) gave a lecture about the Symmetric group.
He demonstrated a short way of calculating the conjugate of a permutation and he talked a bit about S5, including the number of conjugacy classes it has. Each partition of n in Sn represents a conjugacy class. So for 5 it is:
-5 : 24
-41 : 30
-32 : 20
-311 : 20
-221 : 15
-211 : 10
-11111 : 1
Total 120=5!
The Stirling numbers of the first kind are used to calculate these numbers, which incidentally produce a Pascal triangle type-of matrix.
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