

A319643


Number of nonisomorphic weightn strict multiset partitions whose dual is an antichain of (not necessarily distinct) multisets.


1



1, 1, 3, 6, 15, 29, 82, 179, 504, 1302, 3822
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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.
From Gus Wiseman, Aug 15 2019: (Start)
Also the number of nonisomorphic T_0 weak antichains of weight n. The T_0 condition means that the dual is strict (no repeated edges). A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other. For example, nonisomorphic representatives of the a(0) = 1 through a(4) = 15 T_0 weak antichains are:
{} {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1},{1}} {{1,2,2}} {{1,2,2,2}}
{{1},{2}} {{1},{2,2}} {{1,1},{1,1}}
{{1},{1},{1}} {{1,1},{2,2}}
{{1},{2},{2}} {{1},{2,2,2}}
{{1},{2},{3}} {{1,2},{2,2}}
{{1},{2,3,3}}
{{1,3},{2,3}}
{{1},{1},{2,2}}
{{1},{2},{3,3}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
(End)


LINKS

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


EXAMPLE

Nonisomorphic representatives of the a(1) = 1 through a(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
{{1,2}}
{{1},{2}}
3: {{1,1,1}}
{{1,2,3}}
{{1},{1,1}}
{{1},{2,2}}
{{1},{2,3}}
{{1},{2},{3}}
4: {{1,1,1,1}}
{{1,1,2,2}}
{{1,2,3,4}}
{{1},{1,1,1}}
{{1},{1,2,2}}
{{1},{2,2,2}}
{{1},{2,3,4}}
{{1,1},{2,2}}
{{1,2},{3,3}}
{{1,2},{3,4}}
{{1},{2},{1,2}}
{{1},{2},{2,2}}
{{1},{2},{3,3}}
{{1},{2},{3,4}}
{{1},{2},{3},{4}}


CROSSREFS

Cf. A006126, A007716, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616A319646.
Cf. A245567, A293993, A319721, A326704, A326950, A326973, A326978.
Sequence in context: A116696 A000220 A244705 * A092641 A077449 A152232
Adjacent sequences: A319640 A319641 A319642 * A319644 A319645 A319646


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Sep 25 2018


STATUS

approved



