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Number of unlabeled antichains of weight n.
46

%I #17 Aug 18 2019 04:23:43

%S 1,1,2,3,6,9,20,33,72,139

%N Number of unlabeled antichains of weight n.

%C An antichain is a finite set of finite nonempty sets, none of which is a subset of any other. The weight of an antichain is the sum of cardinalities of its elements.

%C From _Gus Wiseman_, Aug 15 2019: (Start)

%C Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n where every vertex is the unique common element of some subset of the edges. For example, the a(1) = 1 through a(6) = 20 set multipartitions are:

%C {1} {1}{1} {1}{1}{1} {1}{2}{12} {1}{2}{2}{12} {12}{13}{23}

%C {1}{2} {1}{2}{2} {1}{1}{1}{1} {1}{2}{3}{23} {1}{2}{12}{12}

%C {1}{2}{3} {1}{1}{2}{2} {1}{1}{1}{1}{1} {1}{2}{13}{23}

%C {1}{2}{2}{2} {1}{1}{2}{2}{2} {1}{2}{3}{123}

%C {1}{2}{3}{3} {1}{2}{2}{2}{2} {1}{1}{2}{2}{12}

%C {1}{2}{3}{4} {1}{2}{2}{3}{3} {1}{1}{2}{3}{23}

%C {1}{2}{3}{3}{3} {1}{2}{2}{2}{12}

%C {1}{2}{3}{4}{4} {1}{2}{3}{3}{23}

%C {1}{2}{3}{4}{5} {1}{2}{3}{4}{34}

%C {1}{1}{1}{1}{1}{1}

%C {1}{1}{1}{2}{2}{2}

%C {1}{1}{2}{2}{2}{2}

%C {1}{1}{2}{2}{3}{3}

%C {1}{2}{2}{2}{2}{2}

%C {1}{2}{2}{3}{3}{3}

%C {1}{2}{3}{3}{3}{3}

%C {1}{2}{3}{3}{4}{4}

%C {1}{2}{3}{4}{4}{4}

%C {1}{2}{3}{4}{5}{5}

%C {1}{2}{3}{4}{5}{6}

%C (End)

%F Euler transform of A293607.

%e Non-isomorphic representatives of the a(5) = 9 antichains are:

%e ((12345)),

%e ((1)(2345)), ((12)(134)), ((12)(345)),

%e ((1)(2)(345)), ((1)(23)(45)), ((2)(13)(14)),

%e ((1)(2)(3)(45)),

%e ((1)(2)(3)(4)(5)).

%Y Cf. A006126, A006602, A007411, A007716, A048143, A049311, A283877, A293607.

%Y Cf. A000372, A000612, A003182, A014466, A055621, A293993, A326704, A326972.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Oct 13 2017