A327129 Number of connected set-systems covering n vertices with at least one edge whose removal (along with any non-covered vertices) disconnects the set-system (non-spanning edge-connectivity 1).

0, 1, 2, 35, 2804

Offset: 0

Author

Gus Wiseman, Aug 27 2019

Keywords

• nonn
• more

Comments

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

Examples

			The a(3) = 35 set-systems:
  {123}  {1}{12}{23}   {1}{2}{12}{13}   {1}{2}{3}{12}{13}
         {1}{13}{23}   {1}{2}{12}{23}   {1}{2}{3}{12}{23}
         {1}{2}{123}   {1}{2}{13}{23}   {1}{2}{3}{13}{23}
         {1}{3}{123}   {1}{2}{3}{123}   {1}{2}{3}{12}{123}
         {2}{12}{13}   {1}{3}{12}{13}   {1}{2}{3}{13}{123}
         {2}{13}{23}   {1}{3}{12}{23}   {1}{2}{3}{23}{123}
         {2}{3}{123}   {1}{3}{13}{23}
         {3}{12}{13}   {2}{3}{12}{13}
         {3}{12}{23}   {2}{3}{12}{23}
         {1}{23}{123}  {2}{3}{13}{23}
         {2}{13}{123}  {1}{2}{13}{123}
         {3}{12}{123}  {1}{2}{23}{123}
                       {1}{3}{12}{123}
                       {1}{3}{23}{123}
                       {2}{3}{12}{123}
                       {2}{3}{13}{123}
		

Crossrefs

The restriction to simple graphs is A327079, with non-covering version A327231.

The version for spanning edge-connectivity is A327145, with BII-numbers A327111.

The BII-numbers of these set-systems are A327099.

The non-covering version is A327196.

Cf. A003465, A006129, A263296, A322395, A323818, A327071, A327149.

Programs

Mathematica

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&eConn[#]==1&]],{n,0,3}]

Formula

Inverse binomial transform of A327196.