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A324461 Number of simple graphs with n vertices and distinct rotations.

%I A324461 #19 Jul 03 2024 14:35:41
%S A324461 1,1,0,6,48,1020,32232,2097144,268369920,68719472640,35184338533920,
%T A324461 36028797018963936,73786976226114539520,302231454903657293676480,
%U A324461 2475880078570197599844819072,40564819207303340847860140736640,1329227995784915854457062986570792960
%N A324461 Number of simple graphs with n vertices and distinct rotations.
%C A324461 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. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).
%H A324461 Andrew Howroyd, <a href="/A324461/b324461.txt">Table of n, a(n) for n = 0..50</a>
%H A324461 Gus Wiseman, <a href="/A324461/a324461.png">The a(4) = 48 graphs with distinct rotations</a>.
%F A324461 a(n > 0) = A306715(n) * n.
%F A324461 a(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
%t A324461 rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
%t A324461 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,0,5}]
%o A324461 (PARI) a(n)={if(n==0, 1, sumdiv(n, d, moebius(d)*2^(n*(n/d-1)/2 + n*(d\2)/d)))} \\ _Andrew Howroyd_, Aug 15 2019
%o A324461 (Python)
%o A324461 from sympy import mobius, divisors
%o A324461 def A324461(n): return sum(mobius(m:=n//d)<<(n*(d-1)>>1)+d*(m>>1) for d in divisors(n,generator=True)) if n else 1 # _Chai Wah Wu_, Jul 03 2024
%Y A324461 Cf. A000088, A000740, A003436, A006125, A027375, A192314, A192332, A306669, A306715, A323860, A323864, A323867, A324462 (covering case), A324463, A324464.
%K A324461 nonn
%O A324461 0,4
%A A324461 _Gus Wiseman_, Feb 28 2019
%E A324461 Terms a(7) and beyond from _Andrew Howroyd_, Aug 15 2019