A293606 Number of unlabeled antichains of weight n.

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

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)

Links

Table of n, a(n) for n=0..9.

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

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

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

Sequence in context: A055873 A246565 A320169 * A185376 A321484 A065536

Adjacent sequences: A293603 A293604 A293605 * A293607 A293608 A293609

Keyword

nonn,more

Author

Gus Wiseman, Oct 13 2017

Status

approved