an:01907100
Zbl 1011.05009
Bender, Edward A.; Canfield, E. Rodney; Richmond, L. Bruce; Wilf, Herbert S.
A discontinuity in the distribution of fixed point sums
EN
Electron. J. Comb. 10, Research paper R15, 18 p. (2003); printed version J. Comb. 10, No. 2 (2003).
00092041
2003
j
05A17 05A20 05A16 11P81
permutation
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.