A cycle is a cyclic sequence of labeled atoms. so that's everything is connected in a circle but now there's fewer of each type that's and in fact, actually if you study it for a minute if you just fix on the largest to smallest element, then you can see that the others are permutation, so actually the counting sequence is N - one factorial. so what's the exponential generating function for cycles? well, it's the sum of N - one factorial C to the N factorial, which everything cancels but an 