A324462 Number of simple graphs covering n vertices with distinct rotations.

1, 0, 0, 3, 28, 765, 26958, 1887277, 252458904, 66376420155, 34508978662350, 35645504882731557, 73356937843604425644, 301275024444053951967585, 2471655539736990372520379226, 40527712706903544100966076156895, 1328579255614092949957261201822704816

Offset

0,4

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. It is covering if there are no isolated vertices. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Gus Wiseman, The a(4) = 28 graph covers with distinct rotations.

Gus Wiseman, The a(4) = 28 graph covers with distinct rotations, radial embedding.

Formula

a(n) = Sum{d|n} mu(n/d) * Sum_{k=0..d} (-1)^(d-k)*binomial(d,k)*2^( k*(k-1)*n/(2*d) + k*(floor(n/(2*d))) ) for n > 0. - Andrew Howroyd, Aug 19 2019

Mathematica

rotgra[g_, m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m, 1, k+1])];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], And[Union@@#==Range[n], UnsameQ@@Table[Nest[rotgra[#, n]&, #, j], {j, n}]]&]], {n, 0, 5}]

Prog

(PARI) a(n)={if(n<1, n==0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, (-1)^(d-k)*binomial(d, k)*2^(k*(k-1)*n/(2*d) + k*(n/d\2)))))} \\ Andrew Howroyd, Aug 19 2019

Crossrefs

Cf. A000088, A000740, A002494, A006125, A006129, A027375, A192332, A323863, A323864, A323867, A323869, A324461 (non-covering case), A324463, A324464.

Sequence in context: A276745 A354664 A015474 * A053601 A328791 A140990

Adjacent sequences: A324459 A324460 A324461 * A324463 A324464 A324465

Keyword

nonn

Author

Gus Wiseman, Feb 28 2019

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 19 2019

Status

approved