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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).
9
0, 1, 2, 35, 2804
OFFSET
0,3
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.
FORMULA
Inverse binomial transform of A327196.
EXAMPLE
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}
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}]
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.
Sequence in context: A110697 A322514 A132100 * A291809 A182070 A325630
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 27 2019
STATUS
approved