A368835 - OEIS

%I #13 Jan 14 2024 16:05:22

%S 0,0,0,1,5,23,98,394,1560,6181,24655,99701,410513,1725725,7423757,

%T 32729320,148027044,687188969,3275077017,16022239940,80431483586,

%U 414094461610,2185052929580,11808696690600,65312048149993,369408792148714,2135111662435080,12601466371445619

%N Number of unlabeled n-edge loop-graphs with at most n vertices such that it is not possible to choose a different vertex from each edge.

%H Andrew Howroyd, <a href="/A368835/b368835.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = A368598(n) - A368984(n). - _Andrew Howroyd_, Jan 14 2024

%e Non-isomorphic representatives of the a(4) = 5 loop-graphs:

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

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

%e   {{1,1},{2,2},{1,2},{3,4}}

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

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

%t Table[Length[Union[sysnorm /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,5}]

%Y The case of a unique choice is A000081, row sums of A106234.

%Y The labeled version is A368596, covering A368730.

%Y Without the choice condition we have A368598.

%Y The complement is A368984, row sums of A368926.

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

%Y A006125 counts graphs, unlabeled A000088.

%Y A006129 counts covering graphs, unlabeled A002494.

%Y A014068 counts loop-graphs, unlabeled A000666.

%Y A058891 counts set-systems (without singletons A016031), unlabeled A000612.

%Y A322661 counts labeled covering half-loop-graphs, connected A062740.

%Y Cf. A116508, A129271, A133686, A137916, A137917, A333331, A367863, A368836, A368924, A368927.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jan 13 2024

%E a(8) onwards from _Andrew Howroyd_, Jan 14 2024