Not sure if this model is natural enough, but if \pi is a random permutation of {1,2..n}, and N_i is the number of cycles of size i in \pi, then, (N_1, N_2...) are approximately independent Poisson(1), Poisson(1/2), Poisson(1/3)... hence probably even if N_i are ranked in order, their sizes would be like 1, 1/2, 1/3 etc.

Not sure if this model is natural enough, but if $\pi$ is a random permutation of $\{1,2..n\}$, and $N_i$ is the number of cycles of size $i$ in $\pi$, then, $(N_1, N_2...)$ are approximately independent Poisson($1$), Poisson($\frac{1}{2}$), Poisson($\frac1{3}$)... hence probably even if $N_i$ are ranked in order, their sizes would be like $1, \frac1{2}, \frac1{3}$ etc.