Geodesic
A discontinuity in the distribution of fixed point sums
The electronic journal of combinatorics, Tome 10 (2003)

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Zbl arXiv EuDML
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.
DOI : 10.37236/1708
Classification : 05A17, 05A20, 05A16, 11P81
Mots-clés : permutation 
Edward A. Bender; E. Rodney Canfield; L. Bruce Richmond; Herbert S. Wilf. A discontinuity in the distribution of fixed point sums. The electronic journal of combinatorics, Tome 10 (2003). doi: 10.37236/1708
@article{10_37236_1708,
     author = {Edward A. Bender and E. Rodney Canfield and L. Bruce Richmond and Herbert S. Wilf},
     title = {A discontinuity in the distribution of fixed point sums},
     journal = {The electronic journal of combinatorics},
     year = {2003},
     volume = {10},
     doi = {10.37236/1708},
     zbl = {1011.05009},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1708/}
}
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