A369147 - OEIS

%I #9 Feb 02 2024 16:11:25

%S 0,0,1,7,52,411,4440,73886,2128608,111533208,10812478194,

%T 1945437194308,650378721118910,404749938336301313,

%U 470163239887698682289,1022592854829028310302180,4177826139658552046624979658,32163829440870460348768017832607,468021728889827507080865185809438918

%N Number of unlabeled loop-graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

%H Andrew Howroyd, <a href="/A369147/b369147.txt">Table of n, a(n) for n = 0..50</a>

%H Gus Wiseman, <a href="/A369147/a369147.png">The a(3) = 7 unlabeled non-choosable loop-graphs</a>.

%F First differences of A369146.

%F a(n) = A322700(n) - A369200(n). - _Andrew Howroyd_, Feb 02 2024

%e The a(0) = 0 through a(3) = 7 loop-graphs (loops shown as singletons):

%e   .  .  {{1},{2},{1,2}}  {{1},{2},{3},{1,2}}

%e                          {{1},{2},{1,2},{1,3}}

%e                          {{1},{2},{1,3},{2,3}}

%e                          {{1},{1,2},{1,3},{2,3}}

%e                          {{1},{2},{3},{1,2},{1,3}}

%e                          {{1},{2},{1,2},{1,3},{2,3}}

%e                          {{1},{2},{3},{1,2},{1,3},{2,3}}

%t 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 Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]]],{n,0,4}]

%Y Without the choice condition we have A322700, labeled A322661.

%Y The complement for exactly n edges is A368984, labeled A333331 (maybe).

%Y The labeled complement is A369140, covering case of A368927.

%Y The labeled version is A369142, covering case of A369141.

%Y This is the covering case of A369146.

%Y The complement is counted by A369200, covering case of A369145.

%Y Without loops we have A369202, covering case of A140637.

%Y A000085, A100861, A111924 count set partitions into singletons or pairs.

%Y A000666 counts unlabeled loop-graphs, labeled A006125 (shifted left).

%Y A002494 counts unlabeled covering graphs, labeled A006129.

%Y A007716 counts non-isomorphic multiset partitions, connected A007718.

%Y Cf. A000088, A000612, A005703, A055621, A062740, A134964, A137917, A140638, A367868, A368835, A369199.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 23 2024

%E a(6) onwards from _Andrew Howroyd_, Feb 02 2024