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A327130
Number of set-systems covering n vertices with spanning edge-connectivity 2.
11
0, 0, 0, 32, 9552
OFFSET
0,4
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.
EXAMPLE
The a(3) = 32 set-systems:
{12}{13}{23} {1}{12}{13}{23} {1}{2}{12}{13}{23} {1}{2}{3}{12}{13}{23}
{12}{13}{123} {2}{12}{13}{23} {1}{3}{12}{13}{23} {1}{2}{3}{12}{13}{123}
{12}{23}{123} {3}{12}{13}{23} {2}{3}{12}{13}{23} {1}{2}{3}{12}{23}{123}
{13}{23}{123} {1}{12}{13}{123} {1}{2}{12}{13}{123} {1}{2}{3}{13}{23}{123}
{1}{12}{23}{123} {1}{2}{12}{23}{123}
{1}{13}{23}{123} {1}{2}{13}{23}{123}
{2}{12}{13}{123} {1}{3}{12}{13}{123}
{2}{12}{23}{123} {1}{3}{12}{23}{123}
{2}{13}{23}{123} {1}{3}{13}{23}{123}
{3}{12}{13}{123} {2}{3}{12}{13}{123}
{3}{12}{23}{123} {2}{3}{12}{23}{123}
{3}{13}{23}{123} {2}{3}{13}{23}{123}
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], #]==2&]], {n, 0, 3}]
CROSSREFS
The BII-numbers of these set-systems are A327108.
Set-systems with spanning edge-connectivity 1 are A327145.
The restriction to simple graphs is A327146.
Sequence in context: A139568 A139294 A227603 * A231035 A213813 A241369
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 27 2019
STATUS
approved