A319642 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is a chain of (not necessarily distinct) multisets.

1, 1, 2, 3, 6, 9, 16, 25, 42, 66, 108

Offset

0,3

Comments

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Links

Table of n, a(n) for n=0..10.

Example

Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1,2},{2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1,2,2,3,3}}
   {{1,2,3,3,3}}
   {{1,2,3,4,4}}
   {{1,2,3,4,5}}
   {{1,2},{2,2,2}}
   {{3,3},{1,2,3}}

Crossrefs

Cf. A000219, A006126, A007716, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646.

Sequence in context: A052812 A213331 A218153 * A062114 A094768 A301753

Adjacent sequences: A319639 A319640 A319641 * A319643 A319644 A319645

Keyword

nonn,more

Author

Gus Wiseman, Sep 25 2018

Status

approved