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 A005326 Permanent of "coprime?" matrix. (Formerly M2382) 7
 1, 1, 3, 4, 28, 16, 256, 324, 3600, 3600, 129744, 63504, 3521232, 3459600, 60891840, 91240704, 8048712960, 3554067456, 425476094976, 320265446400, 12474417291264, 16417666704384, 2778580249611264, 1142807773593600, 172593628397420544 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Number of permutations p of (1,2,3,...,n) such that k and p(k) are relatively prime for all k in (1,2,3,...,n). - Benoit Cloitre, Aug 23 2002 Coprime matrix M=[m(i,j)] is a square matrix defined by m(i,j)=1 if gcd(i,j)=1 else 0. REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Stephen C. Locke, Table of n, a(n) for n = 1..50 (first 30 terms from Seiichi Manyama) D. M. Jackson, The combinatorial interpretation of the Jacobi identity from Lie algebra, J. Combin. Theory, A 23 (1977), 233-256. Carl Pomerance, Coprime permutations, arXiv:2203.03085 [math.NT], 2022. Ashwin Sah and Mehtaab Sawhney, Enumerating coprime permutations, arXiv:2203.06268 [math.NT], 2022. FORMULA a(2n) = A009679(n)^2. - T. D. Noe, Feb 10 2007 MAPLE Jackson2:=proc(n) local m, i, j, M, p, b, s, x; if 0=(n mod 2) then; m := n/2; M := Matrix(m, m, 0); for i from 1 to m do for j from 1 to m do; if 1= igcd(2*i, 2*j-1) then M[i, j]:=1; fi; od; od; s := LinearAlgebra[Permanent](M); return s^2; else; m := (n + 1)/2; M := Matrix(m, m, 0); for i from 1 to m-1 do for j from 1 to m do; if 1=igcd(2*i, 2*j-1) then M[i, j]:=1; fi; od; od; for j to m do M[m, j] := x[j]; end do; p := LinearAlgebra[Permanent](M); b := [ ]; for j to m do b := [op(b), coeff(p, x[j])]; end do; s := 0; for i from 1 to m do for j from 1 to m do; if 1=igcd(2*i-1, 2*j-1) then s:=s+b[i]*b[j]; fi; od; od; fi; return s; end; seq(Jackson2(n), n=1..25); # Stephen C. Locke, Feb 24 2022 MATHEMATICA perm[m_?MatrixQ] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]]; a[n_] := perm[ Table[ Boole[GCD[i, j] == 1], {i, 1, n}, {j, 1, n}]]; Table[an = a[n]; Print[an]; an, {n, 1, 24}] (* Jean-François Alcover, Nov 15 2011 *) (* or, if version >= 10: *) a[n_] := Permanent[Table[Boole[GCD[i, j] == 1], {i, 1, n}, {j, 1, n}]]; Table[an = a[n]; Print[an]; an, {n, 1, 24}] (* Jean-François Alcover, Jul 25 2017 *) PROG (PARI) permRWNb(a)=n=matsize(a)[1]; if(n==1, return(a[1, 1])); sg=1; nc=0; in=vectorv(n); x=in; x=a[, n]-sum(j=1, n, a[, j])/2; p=prod(i=1, n, x[i]); for(k=1, 2^(n-1)-1, sg=-sg; j=valuation(k, 2)+1; z=1-2*in[j]; in[j]+=z; nc+=z; x+=z*a[, j]; p+=prod(i=1, n, x[i], sg)); return(2*(2*(n%2)-1)*p) for(n=1, 26, a=matrix(n, n, i, j, gcd(i, j)==1); print1(permRWNb(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007 CROSSREFS Cf. A009679. Sequence in context: A042829 A232110 A140896 * A298561 A226049 A354844 Adjacent sequences: A005323 A005324 A005325 * A005327 A005328 A005329 KEYWORD nonn,nice AUTHOR N. J. A. Sloane EXTENSIONS Corrected by Vladeta Jovovic, Jul 05 2003 More terms from T. D. Noe, Feb 10 2007 a(25) from Alois P. Heinz, Nov 15 2016 STATUS approved

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Last modified December 5 05:37 EST 2023. Contains 367575 sequences. (Running on oeis4.)