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Any $n$-permutation can by decomposed into the product of disjoint cycles. Assume there are $N_k$ k-cycle, the we can call $(N_1,N_2,\cdots,N_n)$ the type of a given permutation.

My question is how many different types are there, i.e. the number of nonnegative integer solutions of equation $$N_1+2N_2+3N_3+\cdots + nN_n = n$$

Is there a closed form solution? I know how to solve equation in the form of $$N_1+N_2+\cdots + N_k = n$$

But for this question, it is quite unclear to me. Any hints would help. Thanks.

efsdfmo12
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  • This is similar to the "making change for a dollar" problem, see here. Here, $n$ is the amount you want to reach, using coins of values $1, 2, \ldots, n$. – angryavian Nov 28 '23 at 16:55
  • The number of nonnegative integer solutions to the equation $N_1+2N_2+…+nN_n=n$ is equal to the number of integer partitions of $n$. There is no closed form for this number, but there are some complicated infinite summations which yield approximations. – Mike Earnest Nov 28 '23 at 18:01
  • Thank you both for the references. – efsdfmo12 Nov 28 '23 at 19:08

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