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Number of ways of permutations to form a certain pattern of cycles?

Dear all,

How to compute in Mathematica the number of ways of permutations of n-
objects to form a certain pattern of cycles?
I mean, for example, let n = 3. then there are 6 ways of permutations
of three objects

123, 231, 312, 132, 213, 321

Among them, we have the three 1-cycle pattern, 123
two 3-cycle pattern, 231, 312
one 1-cycle and one 2-cycle pattern, 132, 213, 321

How to calculate this kind of mathematical problem with Mathematica?
Could you also show me how to calculate the number of ways without
using of Mathematica?

Thanks so much!

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