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A326572
Number of covering antichains of subsets of {1..n}, all having different sums.
7
2, 1, 2, 8, 80, 3015, 803898
OFFSET
0,1
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}.
EXAMPLE
The a(0) = 2 through a(3) = 8 antichains:
{} {{1}} {{1,2}} {{1,2,3}}
{{}} {{1},{2}} {{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}}
The a(4) = 80 antichains:
{1234} {1}{234} {1}{2}{34} {1}{2}{3}{4} {12}{13}{14}{24}{34}
{12}{34} {1}{3}{24} {1}{23}{24}{34} {12}{13}{23}{24}{34}
{13}{24} {1}{4}{23} {2}{13}{14}{34}
{2}{134} {2}{3}{14} {12}{13}{14}{24}
{3}{124} {1}{23}{24} {12}{13}{14}{34}
{4}{123} {1}{23}{34} {12}{13}{23}{24}
{12}{134} {1}{24}{34} {12}{13}{23}{34}
{12}{234} {2}{13}{14} {12}{13}{24}{34}
{13}{124} {2}{13}{34} {12}{14}{24}{34}
{13}{234} {2}{14}{34} {12}{23}{24}{34}
{14}{123} {3}{14}{24} {13}{14}{24}{34}
{14}{234} {4}{12}{23} {13}{23}{24}{34}
{23}{124} {12}{13}{14} {12}{13}{14}{234}
{23}{134} {12}{13}{24} {12}{23}{24}{134}
{24}{134} {12}{13}{34} {123}{124}{134}{234}
{34}{123} {12}{14}{34}
{123}{124} {12}{23}{24}
{123}{134} {12}{23}{34}
{123}{234} {12}{24}{34}
{124}{134} {13}{14}{24}
{124}{234} {13}{23}{24}
{134}{234} {13}{23}{34}
{13}{24}{34}
{14}{24}{34}
{12}{13}{234}
{12}{14}{234}
{12}{23}{134}
{12}{24}{134}
{13}{14}{234}
{13}{23}{124}
{14}{34}{123}
{23}{24}{134}
{12}{134}{234}
{13}{124}{234}
{14}{123}{234}
{23}{124}{134}
{123}{124}{134}
{123}{124}{234}
{123}{134}{234}
{124}{134}{234}
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}]
CROSSREFS
Antichain covers are A006126.
Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
Antichain covers with equal edge-sums are A326566.
Antichain covers with different edge-sizes are A326570.
The case without singletons is A326571.
Antichains with equal edge-sums are A326574.
Sequence in context: A143208 A353581 A188664 * A119419 A109529 A022694
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 18 2019
STATUS
approved