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A327351
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Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.
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11
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1, 1, 0, 1, 1, 0, 4, 3, 2, 0, 30, 40, 27, 17, 0, 546, 1365, 1842, 1690, 1451, 0, 41334
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OFFSET
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0,7
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COMMENTS
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An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
If empty edges are allowed, we have T(0,0) = 2.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 0
1 1 0
4 3 2 0
30 40 27 17 0
546 1365 1842 1690 1451 0
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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]]]]]]]]];
stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];
vertConnSys[vts_, eds_]:=Min@@Length/@Select[Subsets[vts], Function[del, Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]]
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], Union@@#==Range[n]&&vertConnSys[Range[n], #]==k&]], {n, 0, 4}, {k, 0, n}]
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CROSSREFS
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The version for vertex-connectivity >= k is A327350.
The version for spanning edge-connectivity is A327352.
The version for non-spanning edge-connectivity is A327353, with covering case A327357.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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