A369146 - OEIS

%I #17 Apr 03 2024 03:11:31

%S 0,0,1,8,60,471,4911,78797,2207405,113740613,10926218807,

%T 1956363413115,652335084532025,405402273420833338,

%U 470568642161119515627,1023063423471189429817807,4178849203082023236054797465,32168008290073542372004072630072,468053896898117580623237189882068990

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

%H Andrew Howroyd, <a href="/A369146/b369146.txt">Table of n, a(n) for n = 0..50</a>

%F Partial sums of A369147.

%F a(n) = A000666(n) - A369145(n). - _Andrew Howroyd_, Feb 02 2024

%e The a(0) = 0 through a(3) = 8 loop-graphs (loops shown as singletons):

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

%e                          {{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}]], Select[Tuples[#],UnsameQ@@#&]=={}&]]],{n,0,4}]

%Y Without the choice condition we have A000666, labeled A006125 (shifted).

%Y For a unique choice we have A087803, labeled A088957.

%Y The case without loops is A140637, labeled A367867 (covering A367868).

%Y For exactly n edges we have A368835, labeled A368596.

%Y The labeled complement is A368927, covering A369140.

%Y The labeled version is A369141, covering A369142.

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

%Y The covering case is A369147.

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

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

%Y A129271 counts connected choosable simple graphs, unlabeled A005703.

%Y A322661 counts labeled covering loop-graphs, unlabeled A322700.

%Y Cf. A000088, A000612, A006649, A062740, A134964, A137917, A140638, A368600, A368984, A369199.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 22 2024

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