A Discontinuity in the Distribution of Fixed Point Sums

Abstract

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.