A327145 Number of connected set-systems with n vertices and at least one bridge (spanning edge-connectivity 1).

0, 1, 4, 56, 4640

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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

Crossrefs

The BII-numbers of these set-systems are A327111.

Set systems with non-spanning edge-connectivity 1 are A327196, with covering case A327129.

Set systems with spanning edge-connectivity 2 are A327130.

Cf. A003465, A323818, A327069, A327071, A327099, A327108, A327144.

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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],spanEdgeConn[Range[n],#]==1&]],{n,0,3}]