A369140 Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).

1, 1, 4, 23, 193, 2133, 29410, 486602, 9395315, 207341153, 5147194204, 141939786588, 4304047703755, 142317774817901, 5095781837539766, 196403997108015332, 8106948166404074281, 356781439557643998591, 16675999433772328981216, 824952192369049982670686

Offset: 0

Author

Gus Wiseman, Jan 20 2024

Keywords

• nonn

Comments

These are covering loop-graphs where every connected component has a number of edges less than or equal to the number of vertices in that component. Also covering loop-graphs with at most one cycle (unicyclic) in each connected component.

Examples

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

For a unique choice we have A000272, covering case of A088957.

Without the choice condition we have A322661, unlabeled A322700.

For exactly n edges we have A333331 (maybe), complement A368596.

The case without loops is A367869, covering case of A133686.

This is the covering case of A368927.

The complement is counted by A369142, covering case of A369141.

The unlabeled version is the first differences of A369145.

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

A006125 counts simple graphs; also loop-graphs if shifted left.

A006129 counts covering graphs, unlabeled A002494.

A054548 counts graphs covering n vertices with k edges, with loops A369199.

A367862 counts graphs with n vertices and n edges, covering A367863.

Cf. A000169, A000666, A003465, A006649, A062740, A116508, A137916, A368924, A368984, A369194.

Programs

Mathematica

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

PARI

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(exp(-x + 3*t/2 - 3*t^2/4)/sqrt(1-t) ))} \\ Andrew Howroyd, Feb 02 2024

Formula

Inverse binomial transform of A368927.

Exponential transform of A369197.

E.g.f.: exp(-x)*exp(3*T(x)/2 - 3*T(x)^2/4)/sqrt(1-T(x)), where T(x) is the e.g.f. of A000169. - Andrew Howroyd, Feb 02 2024

Extensions

a(6) onwards from Andrew Howroyd, Feb 02 2024