A318099 -id:A318099 - cp's OEIS frontend

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

Gus Wiseman, Sep 25 2018

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.

			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}}

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

A319645 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is a chain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 9, 16, 22, 38

Offset: 0

Gus Wiseman, Sep 25 2018

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

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 7 antichains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1,2},{2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1,2},{2,2,2}}
6: {{1,1,1,1,1,1}}
   {{1,1,2,2,2,2}}
   {{1,2,2,2,2,2}}
   {{1,2,2,3,3,3}}
   {{1,2},{2,2,2,2}}
   {{1,2,2},{2,2,2}}
   {{1,2,3},{2,3,3}}

Cf. A000219, A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319616-A319646.

cp's OEIS Frontend

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

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