A discontinuity in the distribution of fixed point sums
The electronic journal of combinatorics, Tome 10 (2003)
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.
@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/}
}
TY - JOUR AU - Edward A. Bender AU - E. Rodney Canfield AU - L. Bruce Richmond AU - Herbert S. Wilf TI - A discontinuity in the distribution of fixed point sums JO - The electronic journal of combinatorics PY - 2003 VL - 10 UR - http://geodesic.mathdoc.fr/articles/10.37236/1708/ DO - 10.37236/1708 ID - 10_37236_1708 ER -
%0 Journal Article %A Edward A. Bender %A E. Rodney Canfield %A L. Bruce Richmond %A Herbert S. Wilf %T A discontinuity in the distribution of fixed point sums %J The electronic journal of combinatorics %D 2003 %V 10 %U http://geodesic.mathdoc.fr/articles/10.37236/1708/ %R 10.37236/1708 %F 10_37236_1708
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
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