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A327196
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Number of connected set-systems with n vertices and at least one bridge that is not an endpoint (non-spanning edge-connectivity 1).
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3
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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Non-isomorphic representatives of the a(3) = 44 set-systems:
{{1}}
{{1,2}}
{{1,2,3}}
{{1},{2},{1,2}}
{{1},{1,2},{2,3}}
{{1},{2},{1,2,3}}
{{1},{2,3},{1,2,3}}
{{1},{2},{1,2},{1,3}}
{{1},{2},{1,3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{1},{2},{1,3},{1,2,3}}
{{1},{2},{3},{1,2},{1,3}}
{{1},{2},{3},{1,2},{1,2,3}}
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MATHEMATICA
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csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], 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}]], eConn[#]==1&]], {n, 0, 3}]
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CROSSREFS
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The BII-numbers of these set-systems are A327099.
The restriction to simple graphs is A327231.
Set-systems with spanning edge-connectivity 1 are A327145.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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