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A discontinuity in the distribution of fixed point sums. (English) Zbl 1011.05009
Electron. J. Comb. 10, Research paper R15, 18 p. (2003); printed version J. Comb. 10, No. 2 (2003).
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.
##### MSC:
 05A17 Combinatorial aspects of partitions of integers 05A20 Combinatorial inequalities 05A16 Asymptotic enumeration 11P81 Elementary theory of partitions
permutation
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