A326573 - OEIS

A326573

Number of connected antichains of subsets of {1..n}, all having different sums.

%I #6 Jul 19 2019 07:52:11

%S 1,1,1,5,59,2689,787382

%N Number of connected antichains of subsets of {1..n}, all having different sums.

%C 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 The a(3) = 5 antichains:

%e {{1,2,3}}

%e {{1,3},{2,3}}

%e {{1,2},{2,3}}

%e {{1,2},{1,3}}

%e {{1,2},{1,3},{2,3}}

%e The a(4) = 59 antichains:

%e {1234} {12}{134} {12}{13}{14} {12}{13}{14}{24} {12}{13}{14}{24}{34}

%e {12}{234} {12}{13}{24} {12}{13}{14}{34} {12}{13}{23}{24}{34}

%e {13}{124} {12}{13}{34} {12}{13}{23}{24}

%e {13}{234} {12}{14}{34} {12}{13}{23}{34}

%e {14}{123} {12}{23}{24} {12}{13}{24}{34}

%e {14}{234} {12}{23}{34} {12}{14}{24}{34}

%e {23}{124} {12}{24}{34} {12}{23}{24}{34}

%e {23}{134} {13}{14}{24} {13}{14}{24}{34}

%e {24}{134} {13}{23}{24} {13}{23}{24}{34}

%e {34}{123} {13}{23}{34} {12}{13}{14}{234}

%e {123}{124} {13}{24}{34} {12}{23}{24}{134}

%e {123}{134} {14}{24}{34} {123}{124}{134}{234}

%e {123}{234} {12}{13}{234}

%e {124}{134} {12}{14}{234}

%e {124}{234} {12}{23}{134}

%e {134}{234} {12}{24}{134}

%e {13}{14}{234}

%e {13}{23}{124}

%e {14}{34}{123}

%e {23}{24}{134}

%e {12}{134}{234}

%e {13}{124}{234}

%e {14}{123}{234}

%e {23}{124}{134}

%e {123}{124}{134}

%e {123}{124}{234}

%e {123}{134}{234}

%e {124}{134}{234}

%Y Antichain covers are A006126.

%Y Connected antichains are A048143.

%Y Set partitions with different block-sums are A275780.

%Y MM-numbers of multiset partitions with different part-sums are A326535.

%Y Antichain covers with equal edge-sums are A326566.

%Y The non-connected case is A326572.

%Y Cf. A000372, A293510, A307249, A321469, A323818, A326519, A326565, A326569, A326570, A326571.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Jul 18 2019