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A324462
Number of simple graphs covering n vertices with distinct rotations.
7
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).
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 28 2019
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Aug 19 2019
STATUS
approved