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A326030
Number of antichains of subsets of {1..n} with different edge-sums.
4
2, 3, 6, 19, 132, 3578, 826949
OFFSET
0,1
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}.
EXAMPLE
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}}
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}]
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.
Sequence in context: A121959 A075633 A141048 * A124066 A319204 A093447
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 18 2019
STATUS
approved