This sequence of distributions interests me and I am looking for an expression in closed form. We are looking at populations of size n!. For n=2, we divide the population in half, and assign to each half the property 2 and not-2 (I write 2'), respectively. So S_2 is
S_{2} = \frac{1}{2!}[{1[2],1[2']}].
The 2, 2' are simply attributes of the elements 1,1. For the second,
S_{3} = \frac{1}{3!}[{1[2,3],2[2,3'],2[2'3'],1[2'3]}]
For the fourth,
S_{4} =\frac{1}{4!}[1[2,3,4],3[2,3,4'],2[2,3'4],6[2,3'4'],6[2',3',4'],2[2',3',4],3[2',3,4'],1[2',3,4]...
While the elements ([2,3,4], etc.) can be used to construct the next element of the sequence, I hope the inductive process here is clear.