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If kr<=n, where 1<r<=n. Prove that the number of permutations α ϵ Sn, where α is...

If kr<=n, where 1<r<=n. Prove that the number of permutations α ϵ Sn, where α is a product of k disjoint r-cycles is (1/k!)(1/r^k)[n(n-1)(n-2)...(n-kr+1)]

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Answer #1

Given is a product of k disjoint r-cycles. Hence, looks like

No: of ways in which one can permute kr symbols out of n symbols is .

Each r-cycle can be represented in r-ways . For example, (1234)=(2341)=(3412)=(4123). Hence (1234) can be represented in 4 ways. As there are k disjoint r-cycles, each permutation can be represented in ways.

Permuting k disjoint cycles will not change the permutation. For example (123)(456) =(456)(123). Each permutation can be represented in ways. Combining two arguments each permutation which is a product of k disjoint r-cycles can be represented in ways.

Therefore, No: of k disjoint r-cycles is

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