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A306715
Number of graphical necklaces with n vertices and distinct rotations.
2
1, 0, 2, 12, 204, 5372, 299592, 33546240, 7635496960, 3518433853392, 3275345183542176, 6148914685509544960, 23248573454127484128960, 176848577040728399988915648, 2704321280486889389857342715776, 83076749736557240903566436660674560
OFFSET
1,3
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.
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
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}]
PROG
(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
CROSSREFS
Cf. A000088, A001037, A006125, A059966, A060223, A086675, A192332 (graphical necklaces), A306669, A323861, A323865, A323866, A323871, A324461 (distinct rotations), A324513.
Sequence in context: A317350 A209832 A094157 * A012598 A156489 A129893
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 05 2019
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Aug 15 2019
STATUS
approved