cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A324513 Number of aperiodic cycle necklaces with n vertices.

Original entry on oeis.org

1, 0, 0, 0, 2, 7, 51, 300, 2238, 18028, 164945, 1662067, 18423138, 222380433, 2905942904, 40864642560, 615376173176, 9880203467184, 168483518571789, 3041127459127222, 57926238289894992, 1161157775616335125, 24434798429947993043, 538583682037962702384
Offset: 1

Views

Author

Gus Wiseman, Mar 04 2019

Keywords

Comments

We define an aperiodic cycle necklace to be an equivalence class of (labeled, undirected) Hamiltonian cycles under rotation of the vertices such that all n of these rotations are distinct.

Links

Crossrefs

Cf. A000740, A000939, A001037 (binary Lyndon words), A008965, A059966 (Lyndon compositions), A060223 (normal Lyndon words), A061417, A064852 (if cycle is oriented), A086675, A192332, A275527, A323866 (aperiodic toroidal arrays), A323871.
Cf. A306669, A324461, A324462, A324512, A324514.

Programs

  • Mathematica
    rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1,1]]&/@Permutations[Range[n]]],#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&&UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,8}]
  • PARI
    a(n)={if(n<3, n==0||n==1, (if(n%2, 0, -(n/2-1)!*2^(n/2-2)) + sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2))/2)} \\ Andrew Howroyd, Aug 19 2019

Formula

a(n) = A324512(n)/n.
a(2*n+1) = A064852(2*n+1)/2 for n > 0; a(2*n) = (A064852(2*n) - A002866(n-1))/2 for n > 1. - Andrew Howroyd, Aug 16 2019

Extensions

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

A324514 Number of aperiodic permutations of {1..n}.

Original entry on oeis.org

1, 0, 3, 16, 115, 660, 5033, 39936, 362718, 3624920, 39916789, 478953648, 6227020787, 87177645996, 1307674338105, 20922779566080, 355687428095983, 6402373519409856, 121645100408831981, 2432902004460734000, 51090942171698415483, 1124000727695858073380
Offset: 1

Views

Author

Gus Wiseman, Mar 04 2019

Keywords

Comments

A permutation is defined to be aperiodic if every cyclic rotation of {1..n} acts on the cycle decomposition to produce a different digraph.

Examples

			The a(4) = 16 aperiodic permutations:
  (1243) (1324) (1342) (1423)
  (2134) (2314) (2413) (2431)
  (3124) (3142) (3241) (3421)
  (4132) (4213) (4231) (4312)

Links

Crossrefs

Cf. A000740, A000939, A027375, A061417, A079128, A086675, A192332, A323860, A323867, A323869.
Cf. A306669, A324461, A324462, A324512, A324513, A324514.

Programs

  • Mathematica
    Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]],{n,6}]
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

Formula

a(n) = A306669(n) * n.
a(n) = Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019

Extensions

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

A306669 Number of aperiodic permutation necklaces of weight n.

Original entry on oeis.org

1, 0, 1, 4, 23, 110, 719, 4992, 40302, 362492, 3628799, 39912804, 479001599, 6226974714, 87178289207, 1307673722880, 20922789887999, 355687417744992, 6402373705727999, 121645100223036700, 2432902008176115023, 51090942167993548790, 1124000727777607679999
Offset: 1

Views

Author

Gus Wiseman, Mar 04 2019

Keywords

Comments

A permutation is aperiodic if every rotation of {1...n} acts on the vertices of the cycle decomposition to produce a different digraph. A permutation necklace is an equivalence class of permutations under the action of rotation of vertices in the cycle decomposition. The corresponding action on words applies m -> m + 1 for m < n and n -> 1, and rotates once to the right. For example, (24531) first becomes (35142) under the application of cyclic rotation, and then is rotated right to give (23514).

Links

Crossrefs

Cf. A000031, A000740, A000939, A001037, A059966, A060223, A061417, A086675, A323861, A323865, A323866, A323871.
Cf. A324461, A324512, A324513, A324514.

Programs

  • Mathematica
    Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]]/n,{n,6}]
  • PARI
    a(n) = (1/n)*sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

Formula

a(n) = A324514(n)/n.
a(n) = (1/n)*Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019

Extensions

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019
Showing 1-3 of 3 results.

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