|
%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
| 