A326566 Number of covering antichains of subsets of {1..n} with equal edge-sums.

2, 1, 1, 2, 4, 14, 92, 1320, 73584, 51913039

Offset: 0

Author

Gus Wiseman, Jul 13 2019

Keywords

• nonn
• more

Comments

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

Examples

			The a(1) = 1 through a(5) = 14 antichains:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}      {{1,2,3,4,5}}
                  {{3},{1,2}}  {{1,4},{2,3}}    {{1,2,5},{1,3,4}}
                               {{2,4},{1,2,3}}  {{1,3,5},{2,3,4}}
                               {{3,4},{1,2,4}}  {{1,4,5},{2,3,5}}
                                                {{5},{1,4},{2,3}}
                                                {{1,4,5},{1,2,3,4}}
                                                {{2,3,5},{1,2,3,4}}
                                                {{2,4,5},{1,2,3,5}}
                                                {{3,4,5},{1,2,4,5}}
                                                {{1,5},{2,4},{1,2,3}}
                                                {{2,5},{3,4},{1,2,4}}
                                                {{3,5},{1,2,5},{1,3,4}}
                                                {{4,5},{1,3,5},{2,3,4}}
                                                {{1,4,5},{2,3,5},{1,2,3,4}}
		

Crossrefs

The non-covering case is A326574.

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326518, A326534, A326565.

Programs

Mathematica

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]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n]],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,5}]

Extensions

a(9) from Andrew Howroyd, Aug 14 2019