cp's OEIS Frontend

A368834 Number of unlabeled simple graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).

%I A368834 #11 Feb 02 2024 14:49:55
%S A368834 1,0,1,2,5,10,27,62,165,423,1140,3060,8427,23218,64782,181370,511004,
%T A368834 1444285,4097996,11656644,33243265,94992847,271953126,779790166,
%U A368834 2239187466,6438039076,18532004323,53400606823,154024168401,444646510812,1284682242777
%N A368834 Number of unlabeled simple graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).
%H A368834 Andrew Howroyd, <a href="/A368834/b368834.txt">Table of n, a(n) for n = 0..500</a>
%F A368834 Euler transform of A005703 with A005703(1) = 0.
%F A368834 First differences of A134964.
%F A368834 a(n) = A002494(n) - A369202(n).
%e A368834 Representatives of the a(2) = 1 through a(5) = 10 simple graphs:
%e A368834   {12}  {12}{13}      {12}{34}          {12}{13}{45}
%e A368834         {12}{13}{23}  {12}{13}{14}      {12}{13}{14}{15}
%e A368834                       {12}{13}{24}      {12}{13}{14}{25}
%e A368834                       {12}{13}{14}{23}  {12}{13}{23}{45}
%e A368834                       {12}{13}{24}{34}  {12}{13}{24}{35}
%e A368834                                         {12}{13}{14}{15}{23}
%e A368834                                         {12}{13}{14}{23}{25}
%e A368834                                         {12}{13}{14}{23}{45}
%e A368834                                         {12}{13}{14}{25}{35}
%e A368834                                         {12}{13}{24}{35}{45}
%t A368834 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
%t A368834 Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]],{n,0,5}]
%Y A368834 Without the choice condition we have A002494, labeled A006129.
%Y A368834 The connected case is A005703, labeled A129271.
%Y A368834 This is the covering case of A134964, complement A140637.
%Y A368834 The labeled version is A367869, complement A367868.
%Y A368834 The version with loops is A369200, complement A369147.
%Y A368834 The complement is counted by A369202.
%Y A368834 A007716 counts unlabeled multiset partitions, connected A007718.
%Y A368834 A054548 counts graphs covering n vertices with k edges, with loops A369199.
%Y A368834 A283877 counts unlabeled set-systems, connected A300913.
%Y A368834 Cf. A000088, A006649, A001434, A055621, A116508, A133686, A137916, A137917, A368984, A369146.
%K A368834 nonn
%O A368834 0,4
%A A368834 _Gus Wiseman_, Jan 23 2024