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A327353
Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k.
7
1, 1, 1, 2, 3, 8, 7, 3, 1, 53, 27, 45, 36, 6, 747, 511, 1497, 2085, 1540, 693, 316, 135, 45, 10, 1
OFFSET
0,4
COMMENTS
An antichain is a set of sets, none of which is a subset of any other.
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.
EXAMPLE
Triangle begins:
1
1 1
2 3
8 7 3 1
53 27 45 36 6
747 511 1497 2085 1540 693 316 135 45 10 1
Row n = 3 counts the following antichains:
{} {{1}} {{1,2},{1,3}} {{1,2},{1,3},{2,3}}
{{1},{2}} {{2}} {{1,2},{2,3}}
{{1},{3}} {{3}} {{1,3},{2,3}}
{{2},{3}} {{1,2}}
{{1},{2,3}} {{1,3}}
{{2},{1,3}} {{2,3}}
{{3},{1,2}} {{1,2,3}}
{{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]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1, 0, Length[sys]-Max@@Length/@Select[Union[Subsets[sys]], Length[csm[#]]!=1&]];
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], eConn[#]==k&]], {n, 0, 4}, {k, 0, 2^n}]//.{foe___, 0}:>{foe}
CROSSREFS
Row sums are A014466.
Column k = 0 is A327354.
The covering case is A327357.
The version for spanning edge-connectivity is A327352.
The specialization to simple graphs is A327148, with covering case A327149, unlabeled version A327236, and unlabeled covering case A327201.
Sequence in context: A262992 A036970 A110144 * A270230 A265366 A265365
KEYWORD
nonn,tabf,more
AUTHOR
Gus Wiseman, Sep 10 2019
STATUS
approved