A146557 -id:A146557 - cp's OEIS frontend

A135538 Number of triples in all permutations of order n that are collinear modulo n.

0, 0, 0, 6, 32, 400, 2304, 35280, 322560, 3888000, 48384000, 731808000, 9858723840, 161902540800, 2628760780800, 43181994240000, 876764528640000, 16124496740352000, 358721232629760000, 6933770723303424000, 168738115888742400000, 3644128675321085952000, 94201965756599500800000

Offset: 0

R. J. Mathar, Feb 22 2008, corrected Oct 24 2008

nonn

			For example, in a permutation p=[5,2,4,3,1], a triple of points { (2,p(2)=2), (4,p(4)=3), (5,p(5)=1) } is collinear, since they are located on the line: x + 3*y == 3 (mod 5).

Alois P. Heinz, Table of n, a(n) for n = 0..400
L. Li, Collinear triples in permutations, arXiv:0802.0572 [math.CO], 2008.

f[n_] := n*Sum[ Sum[ (n - i - j)*( n*GCD[i, j, n - i - j] - GCD[i, n] - GCD[j, n] - GCD[i + j, n] + 2 ) , {j, 1, n - i}] , {i, 1, n}]; Join[{0, 0, 0}, Table[(n - 3)!*f[n], {n, 3, 25}]] (* G. C. Greubel, Oct 18 2016 *)

a(n) = if(n<3, 0, (n-3)! * n * sum(i=1,n, sum(j=1,n-i-1, (n-i-j) * (n*gcd([i,j,n-i-j]) - gcd(i,n) - gcd(j,n) - gcd(i+j,n) + 2) )))

For n>=3, a(n) = (n-3)! * A146557(n).

Edited and extended by Max Alekseyev, Oct 31 2008

A146558 Number of order n permutations without collinear triples modulo n.

Original entry on oeis.org

1, 2, 0, 16, 0, 72, 0, 256, 0, 0, 0, 2304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Max Alekseyev, Nov 01 2008

			For n=4, there are a(4)=16 permutations without collinear triples: [1, 2, 4, 3], [1, 3, 2, 4], [1, 3, 4, 2], [1, 4, 2, 3], [2, 1, 3, 4], [2, 3, 1, 4], [2, 4, 1, 3], [2, 4, 3, 1], [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 4, 1], [3, 4, 2, 1], [4, 1, 3, 2], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2]

L. Li, Collinear triples in permutations, arXiv:0802.0572 [math.CO], 2008.

Cf. A135538, A146557.

{ a(n) = local(p,r,g); r=0; for(j=1,n!, p=numtoperm(n,j); g=1; forvec(v=vector(3,i,[1,n]), if(matdet([1,v[1],p[v[1]];1,v[2],p[v[2]];1,v[3],p[v[3]]])%n==0, g=0; break), 2); if(g,r++)); r }

For prime p>=3, a(p) = 0.

Edited by Max Alekseyev, Jun 21 2010

a(14)-a(29) from Bert Dobbelaere, Mar 15 2020

cp's OEIS Frontend

A135538 Number of triples in all permutations of order n that are collinear modulo n.

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