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.