A327227 Number of labeled simple graphs covering n vertices with at least one endpoint/leaf.

0, 0, 1, 3, 31, 515, 15381, 834491, 83016613, 15330074139, 5324658838645, 3522941267488973, 4489497643961740521, 11119309286377621015089, 53893949089393110881259181, 513788884660608277842596504415, 9669175277199248753133328740702449

Offset: 0

Author

Gus Wiseman, Sep 01 2019

Keywords

• nonn

Comments

Covering means there are no isolated vertices.

A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also graphs with minimum vertex-degree 1.

Examples

			The a(4) = 31 edge-sets:
  {12,34}  {12,13,14}  {12,13,14,23}
  {13,24}  {12,13,24}  {12,13,14,24}
  {14,23}  {12,13,34}  {12,13,14,34}
           {12,14,23}  {12,13,23,24}
           {12,14,34}  {12,13,23,34}
           {12,23,24}  {12,14,23,24}
           {12,23,34}  {12,14,24,34}
           {12,24,34}  {12,23,24,34}
           {13,14,23}  {13,14,23,34}
           {13,14,24}  {13,14,24,34}
           {13,23,24}  {13,23,24,34}
           {13,23,34}  {14,23,24,34}
           {13,24,34}
           {14,23,24}
           {14,23,34}
           {14,24,34}
		

Links

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

Crossrefs

Column k=1 of A327366.

The non-covering version is A245797.

The unlabeled version is A324693.

The generalization to set-systems is A327229.

BII-numbers of set-systems with minimum degree 1 are A327105.

Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327228, A327230.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

Formula

Inverse binomial transform of A245797, if we assume A245797(0) = 0.