On the Number of Collinear Triples in Permutations
نویسنده
چکیده
Let α : Zn → Zn be a permutation and Ψ(α) be the number of collinear triples modulo n in the graph of α. Cooper and Solymosi had given by induction the bound minα Ψ(α) ≥ ⌈(n − 1)/4⌉ when n is a prime number. The main purpose of this paper is to give a direct proof of that bound. Besides, the expected number of collinear triples a permutation can have is also been determined.
منابع مشابه
Collinear Triples in Permutations
Let α : Fq → Fq be a permutation and Ψ(α) be the number of collinear triples in the graph of α, where Fq denotes a finite field of q elements. When q is odd Cooper and Solymosi once proved Ψ(α) ≥ (q − 1)/4 and conjectured the sharp bound should be Ψ(α) ≥ (q−1)/2. In this note we indicate that the Cooper-Solymosi conjecture is true.
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