OFFSET
0,3
COMMENTS
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.
From Gus Wiseman, Aug 15 2019: (Start)
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:
{1} {1}{1} {1}{1}{1} {1}{2}{12} {1}{2}{2}{12} {12}{13}{23}
{1}{2} {1}{2}{2} {1}{1}{1}{1} {1}{2}{3}{23} {1}{2}{12}{12}
{1}{2}{3} {1}{1}{2}{2} {1}{1}{1}{1}{1} {1}{2}{13}{23}
{1}{2}{2}{2} {1}{1}{2}{2}{2} {1}{2}{3}{123}
{1}{2}{3}{3} {1}{2}{2}{2}{2} {1}{1}{2}{2}{12}
{1}{2}{3}{4} {1}{2}{2}{3}{3} {1}{1}{2}{3}{23}
{1}{2}{3}{3}{3} {1}{2}{2}{2}{12}
{1}{2}{3}{4}{4} {1}{2}{3}{3}{23}
{1}{2}{3}{4}{5} {1}{2}{3}{4}{34}
{1}{1}{1}{1}{1}{1}
{1}{1}{1}{2}{2}{2}
{1}{1}{2}{2}{2}{2}
{1}{1}{2}{2}{3}{3}
{1}{2}{2}{2}{2}{2}
{1}{2}{2}{3}{3}{3}
{1}{2}{3}{3}{3}{3}
{1}{2}{3}{3}{4}{4}
{1}{2}{3}{4}{4}{4}
{1}{2}{3}{4}{5}{5}
{1}{2}{3}{4}{5}{6}
(End)
FORMULA
Euler transform of A293607.
EXAMPLE
Non-isomorphic representatives of the a(5) = 9 antichains are:
((12345)),
((1)(2345)), ((12)(134)), ((12)(345)),
((1)(2)(345)), ((1)(23)(45)), ((2)(13)(14)),
((1)(2)(3)(45)),
((1)(2)(3)(4)(5)).
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 13 2017
STATUS
approved