A135616 Number of permutations p of {1,2,...,n} such that p(x) is a polynomial in x, modulo n, of degree at most 2, for x=1,2,3,...,n.
1, 2, 6, 8, 20, 12, 42, 64, 162, 40, 110, 48, 156, 84, 120, 512, 272, 324, 342, 160, 252, 220, 506, 384, 2500, 312, 4374, 336, 812, 240, 930, 4096, 660, 544, 840, 1296, 1332, 684, 936, 1280
Offset: 1
Keywords
Examples
For n=4, the permutation (1,2,3,4) is clearly given by the polynomial p(x)=x, for any modulus and the permutation (1,4,3,2) is found to be given by p(x)=2x^2+x+2 (modulo 4), since 2+1+2=5=1(mod 4), 2*4+2+2=12=0 (mod 4), 2*9+3+2=23=3 (mod 4) and 2*16+4+2=38=2 (mod 4). Among the other 22 permutations of (1,2,3,4) six are found to have the desired property, for a total of 8, so a(4)=8.
Crossrefs
Cf. A002618 (analog with linear polynomials).
Programs
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Mathematica
f = Function[n, arg = Range[n]; Length[Union[Select[Flatten[ Table[Mod[a*arg^2 + b*arg + c, n], {a, n}, {b, n}, {c, n}], 2], Sort[#] == arg - 1 &]]]]; Table[f[n], {n, 40}] (* Ivan Neretin, May 21 2015 *) -
PARI
a(n)=my(u=List(),v); for(a=1,n-1,for(b=0,n-1, v=vector(n,x,a*x^2+b*x)%n; if(#Set(v)==n, listput(u,v)))); for(a=1,n, v=vector(n,x,a*x%n); if(#Set(v)==n,listput(u,v))); n*#Set(u) \\ Charles R Greathouse IV, May 21 2015
Extensions
More terms from Ivan Neretin, May 21 2015
Comments