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A327351 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k. 11
1, 1, 0, 1, 1, 0, 4, 3, 2, 0, 30, 40, 27, 17, 0, 546, 1365, 1842, 1690, 1451, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

Triangle begins:

    1

    1    0

    1    1    0

    4    3    2    0

   30   40   27   17    0

  546 1365 1842 1690 1451    0

MATHEMATICA

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}]

CROSSREFS

Row sums are A307249, or A006126 if empty edges are allowed.

Column k = 0 is A120338, if we assume A120338(0) = A120338(1) = 1.

Column k = 1 is A327356.

Column k = n - 1 is A327020.

The unlabeled version is A327359.

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.

Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336.

Sequence in context: A258692 A067018 A200233 * A239475 A100802 A022960

Adjacent sequences:  A327348 A327349 A327350 * A327352 A327353 A327354

KEYWORD

nonn,tabl,more

AUTHOR

Gus Wiseman, Sep 09 2019

STATUS

approved

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Last modified November 21 16:04 EST 2019. Contains 329371 sequences. (Running on oeis4.)