OFFSET
0,4
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(3) = 5 antichains:
{{1,2,3}}
{{1,3},{2,3}}
{{1,2},{2,3}}
{{1,2},{1,3}}
{{1,2},{1,3},{2,3}}
The a(4) = 61 antichains:
{1234} {12}{34} {12}{13}{14} {12}{13}{14}{24} {12}{13}{14}{24}{34}
{13}{24} {12}{13}{24} {12}{13}{14}{34} {12}{13}{23}{24}{34}
{12}{134} {12}{13}{34} {12}{13}{23}{24}
{12}{234} {12}{14}{34} {12}{13}{23}{34}
{13}{124} {12}{23}{24} {12}{13}{24}{34}
{13}{234} {12}{23}{34} {12}{14}{24}{34}
{14}{123} {12}{24}{34} {12}{23}{24}{34}
{14}{234} {13}{14}{24} {13}{14}{24}{34}
{23}{124} {13}{23}{24} {13}{23}{24}{34}
{23}{134} {13}{23}{34} {12}{13}{14}{234}
{24}{134} {13}{24}{34} {12}{23}{24}{134}
{34}{123} {14}{24}{34} {123}{124}{134}{234}
{123}{124} {12}{13}{234}
{123}{134} {12}{14}{234}
{123}{234} {12}{23}{134}
{124}{134} {12}{24}{134}
{124}{234} {13}{14}{234}
{134}{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], {2, 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 and no singletons are A326565.
Antichain covers with different edge-sizes and no singletons are A326569.
The case with singletons allowed is A326572.
Antichains with equal edge-sums are A326574.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 18 2019
STATUS
approved