OFFSET
0,7
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.
FORMULA
Euler transform of A319625.
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(10) = 7 antichains:
1: {{1}}
2: {{1},{2}}
3: {{1},{2},{3}}
4: {{1},{2},{3},{4}}
5: {{1},{2},{3},{4},{5}}
6: {{1,2},{1,3},{2,3}}
{{1},{2},{3},{4},{5},{6}}
7: {{1},{2,3},{2,4},{3,4}}
{{1},{2},{3},{4},{5},{6},{7}}
8: {{1,2},{1,3},{2,4},{3,4}}
{{1},{2},{3,4},{3,5},{4,5}}
{{1},{2},{3},{4},{5},{6},{7},{8}}
9: {{1,2},{1,3},{1,4},{2,3,4}}
{{1},{2,3},{2,4},{3,5},{4,5}}
{{1},{2},{3},{4,5},{4,6},{5,6}}
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
10: {{1,3},{2,4},{1,2,5},{3,4,5}}
{{1},{2,3},{2,4},{2,5},{3,4,5}}
{{1,2},{1,3},{2,4},{3,5},{4,5}}
{{1,3},{1,4},{2,3},{2,4},{3,4}}
{{1},{2},{3,4},{3,5},{4,6},{5,6}}
{{1},{2},{3},{4},{5,6},{5,7},{6,7}}
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 25 2018
STATUS
approved