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%I A326573 #6 Jul 19 2019 07:52:11
%S A326573 1,1,1,5,59,2689,787382
%N A326573 Number of connected antichains of subsets of {1..n}, all having different sums.
%C A326573 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}.
%e A326573 The a(3) = 5 antichains:
%e A326573 {{1,2,3}}
%e A326573 {{1,3},{2,3}}
%e A326573 {{1,2},{2,3}}
%e A326573 {{1,2},{1,3}}
%e A326573 {{1,2},{1,3},{2,3}}
%e A326573 The a(4) = 59 antichains:
%e A326573 {1234} {12}{134} {12}{13}{14} {12}{13}{14}{24} {12}{13}{14}{24}{34}
%e A326573 {12}{234} {12}{13}{24} {12}{13}{14}{34} {12}{13}{23}{24}{34}
%e A326573 {13}{124} {12}{13}{34} {12}{13}{23}{24}
%e A326573 {13}{234} {12}{14}{34} {12}{13}{23}{34}
%e A326573 {14}{123} {12}{23}{24} {12}{13}{24}{34}
%e A326573 {14}{234} {12}{23}{34} {12}{14}{24}{34}
%e A326573 {23}{124} {12}{24}{34} {12}{23}{24}{34}
%e A326573 {23}{134} {13}{14}{24} {13}{14}{24}{34}
%e A326573 {24}{134} {13}{23}{24} {13}{23}{24}{34}
%e A326573 {34}{123} {13}{23}{34} {12}{13}{14}{234}
%e A326573 {123}{124} {13}{24}{34} {12}{23}{24}{134}
%e A326573 {123}{134} {14}{24}{34} {123}{124}{134}{234}
%e A326573 {123}{234} {12}{13}{234}
%e A326573 {124}{134} {12}{14}{234}
%e A326573 {124}{234} {12}{23}{134}
%e A326573 {134}{234} {12}{24}{134}
%e A326573 {13}{14}{234}
%e A326573 {13}{23}{124}
%e A326573 {14}{34}{123}
%e A326573 {23}{24}{134}
%e A326573 {12}{134}{234}
%e A326573 {13}{124}{234}
%e A326573 {14}{123}{234}
%e A326573 {23}{124}{134}
%e A326573 {123}{124}{134}
%e A326573 {123}{124}{234}
%e A326573 {123}{134}{234}
%e A326573 {124}{134}{234}
%Y A326573 Antichain covers are A006126.
%Y A326573 Connected antichains are A048143.
%Y A326573 Set partitions with different block-sums are A275780.
%Y A326573 MM-numbers of multiset partitions with different part-sums are A326535.
%Y A326573 Antichain covers with equal edge-sums are A326566.
%Y A326573 The non-connected case is A326572.
%Y A326573 Cf. A000372, A293510, A307249, A321469, A323818, A326519, A326565, A326569, A326570, A326571.
%K A326573 nonn,more
%O A326573 0,4
%A A326573 _Gus Wiseman_, Jul 18 2019