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A327352
Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.
8
1, 1, 1, 4, 1, 14, 4, 1, 83, 59, 23, 2, 1232, 2551, 2792, 887, 107, 10, 1
OFFSET
0,4
COMMENTS
An antichain is a set of sets, none of which is a subset of any other.
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 set-system that is disconnected or covers fewer vertices.
EXAMPLE
Triangle begins:
1
1 1
4 1
14 4 1
83 59 23 2
1232 2551 2792 887 107 10 1
Row n = 3 counts the following antichains:
{} {{1,2,3}} {{1,2},{1,3},{2,3}}
{{1}} {{1,2},{1,3}}
{{2}} {{1,2},{2,3}}
{{3}} {{1,3},{2,3}}
{{1,2}}
{{1,3}}
{{2,3}}
{{1},{2}}
{{1},{3}}
{{2},{3}}
{{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{2},{3}}
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]]]];
spanEdgeConn[vts_, eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds], Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], spanEdgeConn[Range[n], #]==k&]], {n, 0, 4}, {k, 0, 2^n}]//.{foe___, 0}:>{foe}
CROSSREFS
Row sums are A014466.
Column k = 0 is A327355.
The unlabeled version is A327438.
Sequence in context: A347486 A335337 A226906 * A050156 A191584 A231185
KEYWORD
nonn,tabf,more
AUTHOR
Gus Wiseman, Sep 10 2019
STATUS
approved