A368835 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.

0, 0, 0, 1, 5, 23, 98, 394, 1560, 6181, 24655, 99701, 410513, 1725725, 7423757, 32729320, 148027044, 687188969, 3275077017, 16022239940, 80431483586, 414094461610, 2185052929580, 11808696690600, 65312048149993, 369408792148714, 2135111662435080, 12601466371445619

Offset: 0

Author

Gus Wiseman, Jan 13 2024

Keywords

• nonn

Examples

			Non-isomorphic representatives of the a(4) = 5 loop-graphs:
  {{1,1},{2,2},{3,3},{1,2}}
  {{1,1},{2,2},{1,2},{1,3}}
  {{1,1},{2,2},{1,2},{3,4}}
  {{1,1},{2,2},{1,3},{2,3}}
  {{1,1},{1,2},{1,3},{2,3}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

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

The labeled version is A368596, covering A368730.

Without the choice condition we have A368598.

The complement is A368984, row sums of A368926.

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

A006125 counts graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

A014068 counts loop-graphs, unlabeled A000666.

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

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

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

Programs

Mathematica

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

Formula

a(n) = A368598(n) - A368984(n). - Andrew Howroyd, Jan 14 2024

Extensions

a(8) onwards from Andrew Howroyd, Jan 14 2024