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A319643 Number of non-isomorphic weight-n 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 (list; graph; refs; listen; history; text; internal format)
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 non-isomorphic 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, non-isomorphic 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

Non-isomorphic 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, A319616-A319646.

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

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Last modified December 6 14:15 EST 2019. Contains 329806 sequences. (Running on oeis4.)