A306715 Number of graphical necklaces with n vertices and distinct rotations.

1, 0, 2, 12, 204, 5372, 299592, 33546240, 7635496960, 3518433853392, 3275345183542176, 6148914685509544960, 23248573454127484128960, 176848577040728399988915648, 2704321280486889389857342715776, 83076749736557240903566436660674560

Offset: 1

Author

Gus Wiseman, Mar 05 2019

Keywords

• nonn

Comments

A simple graph with n vertices has distinct rotations if all n rotations of its vertex set act on the edge set to give distinct graphs. A graphical necklace is a simple graph that is minimal among all n rotations of the vertices.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Gus Wiseman, Inequivalent representatives of the a(4) = 12 graphical necklaces with distinct rotations.

Crossrefs

Cf. A000088, A001037, A006125, A059966, A060223, A086675, A192332 (graphical necklaces), A306669, A323861, A323865, A323866, A323871, A324461 (distinct rotations), A324513.

Programs

Mathematica

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],With[{rots=Table[Nest[rotgra[#,n]&,#,j],{j,n}]},UnsameQ@@rots&&#==First[Sort[rots]]]&]],{n,5}]

PARI

a(n)={if(n==0, 1, sumdiv(n, d, moebius(d)*2^(n*(n/d-1)/2 + n*(d\2)/d))/n)} \\ Andrew Howroyd, Aug 15 2019

Formula

a(n > 0) = A324461(n)/n.

a(n) = (1/n)*Sum_{d|n} mu(d)*2^(n*(n/d-1)/2 + n*floor(d/2)/d) for n > 0. - Andrew Howroyd, Aug 15 2019

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 15 2019