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.

1, 1, 1, 4, 1, 14, 4, 1, 83, 59, 23, 2, 1232, 2551, 2792, 887, 107, 10, 1

Offset: 0

Author

Gus Wiseman, Sep 10 2019

Keywords

• nonn
• tabf
• more

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.

Examples

			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}}
		

Crossrefs

Row sums are A014466.

Column k = 0 is A327355.

The unlabeled version is A327438.

Cf. A052446, A327062, A327071, A327103, A327111, A327144, A327351, A327353.

Programs

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}