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2, 3, 5, 10, 22, 61, 247, 2096, 81896, 52260575

a(9) from Andrew Howroyd, Aug 13 2019

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allocated for Gus WisemanNumber of antichains of subsets of {1..n} with equal edge-sums.

2, 3, 5, 10, 22, 61, 247, 2096, 81896

0,1

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}.

The a(0) = 2 through a(4) = 22 antichains:

{} {} {} {} {}

{{}} {{}} {{}} {{}} {{}}

{{1}} {{1}} {{1}} {{1}}

{{2}} {{2}} {{2}}

{{1,2}} {{3}} {{3}}

{{1,2}} {{4}}

{{1,3}} {{1,2}}

{{2,3}} {{1,3}}

{{1,2,3}} {{1,4}}

{{3},{1,2}} {{2,3}}

{{2,4}}

{{3,4}}

{{1,2,3}}

{{1,2,4}}

{{1,3,4}}

{{2,3,4}}

{{1,2,3,4}}

{{3},{1,2}}

{{4},{1,3}}

{{1,4},{2,3}}

{{2,4},{1,2,3}}

{{3,4},{1,2,4}}

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}]

Set partitions with equal block-sums are A035470.

Antichains with different edge-sums are A326030.

MM-numbers of multiset partitions with equal part-sums are A326534.

The covering case is A326566.

Cf. A000372, A003182, A006126, A307249, A321455, A321717, A321718, A326518, A326565, A326572.

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Gus Wiseman, Jul 18 2019

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allocated for Gus Wiseman

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