  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.
