A326030 Number of antichains of subsets of {1..n} with different edge-sums.

2, 3, 6, 19, 132, 3578, 826949

Offset: 0

Author

Gus Wiseman, Jul 18 2019

Keywords

• nonn
• more

Comments

An antichain is a finite set of finite sets, none of which is a subset of any other. 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(0) = 2 through a(3) = 19 antichains:
  {}    {}     {}         {}
  {{}}  {{}}   {{}}       {{}}
        {{1}}  {{1}}      {{1}}
               {{2}}      {{2}}
               {{1,2}}    {{3}}
               {{1},{2}}  {{1,2}}
                          {{1,3}}
                          {{2,3}}
                          {{1},{2}}
                          {{1,2,3}}
                          {{1},{3}}
                          {{2},{3}}
                          {{1},{2,3}}
                          {{2},{1,3}}
                          {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}
		

Crossrefs

Set partitions with different block-sums are A275780.

MM-numbers of multiset partitions with different part-sums are A326535.

The covering case is A326572.

Antichains with equal edge-sums are A326574.

Cf. A000372, A003182, A006126, A321469, A326519, A326571, A326573.

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]]]];
cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]==Total[#2]&];
Table[Length[cleqset[Range[n]]],{n,0,5}]