Compute the number of conjugacy classes of elements of prime order in Sym(n)

Let p be a prime. Find a formula for the number of conjugacy classes of elements of order p in S_n using the floor function.


Every element of order p in S_n is a product of commuting p-cycles. Provided mp \leq n, there is a conjugacy class in S_n consisting of all products of m commuting p-cycles; that is, one class for all integers 1\leq m \leq n/p. Thus the number of such classes is \lfloor n/p \rfloor.

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