A discontinuity in the distribution of fixed point sums (Q1871379)

Summary: The quantity \(f(n,r)\), defined as the number of permutations of the set \([n]=\{1,2,\dots,n\}\) 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.