不动点和分布的一个间断
A Discontinuity in the Distribution of Fixed Point Sums
摘要：
数量$f(n,r)$，定义为集合$[n]=\{1,2,... n\}$的排列数，其不动点之和为$r$，在$r=n$的邻域中显示出尖锐的不连续性。我们解释这种不连续性和研究可能存在的其他不连续性$f(n,r)$的排列。通过研究$f(n,r)$，当“不动点"在结构的适当图中被”大小为1的分量“取代时，我们将我们的结果推广到其他具有相同不连续性的结构族。其中考虑的对象是排列，所有的功能和集分区。
英文摘要：
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.
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