The quantity f(n,r), defined as the number of permutations of the set
[n]={1,2,...n} whose fixed points sum to r, shows a sharp discontinuity
in the neighborhood of r=n. We explain this discontinuity and study the
possible existence of other discontinuities in f(n,r) for permutations. We
generalize our results to other families of structures that exhibit the same
kind of discontinuities, by studying f(n,r) when ``fixed points'' is replaced
by ``components of size 1'' in a suitable graph of the structure. Among the
objects considered are permutations, all functions and set partitions.Comment: 1 figur