A319557 -id:A319557 - cp's OEIS frontend

A319628 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.

1, 1, 2, 2, 3, 3, 10, 11, 37, 80, 233

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(6) = 10 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
3: {{1,1,1}}
   {{1,2,3}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,3,4}}
5: {{1,1,1,1,1}}
   {{1,2,3,4,5}}
   {{1,1},{1,2,2}}
6: {{1,1,1,1,1,1}}
   {{1,1,1,2,2,2}}
   {{1,1,2,2,3,3}}
   {{1,2,3,4,5,6}}
   {{1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2}}
   {{1,1,2},{2,2,2}}
   {{1,1,2},{2,3,3}}
   {{1,1},{1,2},{2,2}}
   {{1,2},{1,3},{2,3}}

Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646.

Euler transform is A319641.

A319793 Number of non-isomorphic connected strict multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 24, 96, 412, 1607, 6348, 24580, 96334, 378569, 1508220, 6079720, 24879878, 103335386, 436032901, 1869019800, 8139613977, 36008825317, 161794412893, 738167013847, 3418757243139, 16068569129711, 76622168743677, 370571105669576, 1817199912384794

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

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(4) = 1 through a(5) = 4 multiset partitions:
4:  {{1},{2},{1,2}}
5: {{1},{2},{1,2,2}}
   {{1},{1,2},{2,2}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A007718, A056156, A281116, A283877, A316980, A317752, A317755, A317757, A319616.

Cf. A319077, A319748, A319755, A319778, A319781, A319791.

a(n) = A319557(n) - A316980(n) + A319077(n). - Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

A319619 Number of non-isomorphic connected weight-n antichains of multisets whose dual is also an antichain of multisets.

Original entry on oeis.org

1, 1, 3, 3, 6, 4, 15, 13, 48, 96, 280

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) = 4 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,3}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,3,4}}
   {{1,1},{1,1}}
   {{1,2},{1,2}}
   {{1},{1},{1},{1}}
5: {{1,1,1,1,1}}
   {{1,2,3,4,5}}
   {{1,1},{1,2,2}}
   {{1},{1},{1},{1},{1}}

Cf. A006126, A007716, A007718, A056156, A059201, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646, A300913.

Euler transform is A318099.

A319620 Number of connected antichain covers of n vertices by distinct sets whose dual is also a (not necessarily strict) antichain.

Original entry on oeis.org

1, 1, 1, 2, 22, 2133

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 a(4) = 22 antichain covers:
   {{1,2,3,4}}
   {{3,4},{1,2,3},{1,2,4}}
   {{2,4},{1,2,3},{1,3,4}}
   {{2,3},{1,2,4},{1,3,4}}
   {{1,4},{1,2,3},{2,3,4}}
   {{1,3},{1,2,4},{2,3,4}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3},{2,4}}
   {{1,2},{1,4},{2,3},{3,4}}
   {{1,2},{1,3},{2,4},{3,4}}
   {{1,4},{2,4},{3,4},{1,2,3}}
   {{1,3},{2,3},{3,4},{1,2,4}}
   {{1,2},{2,3},{2,4},{1,3,4}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,3},{1,4},{2,3},{2,4},{3,4}}
   {{1,2},{1,4},{2,3},{2,4},{3,4}}
   {{1,2},{1,3},{2,3},{2,4},{3,4}}
   {{1,2},{1,3},{1,4},{2,4},{3,4}}
   {{1,2},{1,3},{1,4},{2,3},{3,4}}
   {{1,2},{1,3},{1,4},{2,3},{2,4}}
   {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
   {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A006126, A007716, A007718, A049311, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319565, A319616-A319646, A300913.

A319621 Number of non-isomorphic connected antichain covers of n vertices by distinct sets whose dual is also an antichain of (not necessarily distinct) sets.

Original entry on oeis.org

1, 1, 1, 2, 7, 73

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}}.

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

Cf. A006126, A007716, A007718, A049311, A056156, A059201, A316980, A316983, A318099, A319557, A319565, A319616-A319646, A300913.

A319622 Number of non-isomorphic connected weight-n antichains of distinct sets whose dual is also an antichain of (not necessarily distinct) sets.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 7

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(10) = 7 antichains:
1:  {{1}}
2:  {{1,2}}
3:  {{1,2,3}}
4:  {{1,2,3,4}}
5:  {{1,2,3,4,5}}
6:  {{1,2,3,4,5,6}}
    {{1,2},{1,3},{2,3}}
7:  {{1,2,3,4,5,6,7}}
8:  {{1,2,3,4,5,6,7,8}}
    {{1,2},{1,3,4},{2,3,4}}
    {{1,2},{1,3},{2,4},{3,4}}
9:  {{1,2,3,4,5,6,7,8,9}}
    {{1,2},{1,3},{1,4},{2,3,4}}
10: {{1,2,3,4,5,6,7,8,9,10}}
    {{1,2},{1,3,4,5},{2,3,4,5}}
    {{1,2,3},{1,4,5},{2,3,4,5}}
    {{1,2},{1,3},{2,4,5},{3,4,5}}
    {{1,3},{2,4},{1,2,5},{3,4,5}}
    {{1,2},{1,3},{2,4},{3,5},{4,5}}
    {{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319565, A319616-A319646, A300913.

A319623 Number of connected antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

Original entry on oeis.org

1, 1, 0, 1, 15, 1957

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}}.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 antichain covers:
1: {{1}}
3: {{1,2},{1,3},{2,3}}
4: {{1,2},{1,3},{2,4},{3,4}}
   {{1,3},{1,4},{2,3},{2,4}}
   {{1,2},{1,4},{2,3},{3,4}}
   {{1,4},{2,4},{3,4},{1,2,3}}
   {{1,3},{2,3},{3,4},{1,2,4}}
   {{1,2},{2,3},{2,4},{1,3,4}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,3},{1,4},{2,3},{2,4},{3,4}}
   {{1,2},{1,4},{2,3},{2,4},{3,4}}
   {{1,2},{1,3},{2,3},{2,4},{3,4}}
   {{1,2},{1,3},{1,4},{2,4},{3,4}}
   {{1,2},{1,3},{1,4},{2,3},{3,4}}
   {{1,2},{1,3},{1,4},{2,3},{2,4}}
   {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
   {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646, A300913.

A319624 Number of non-isomorphic connected antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

Original entry on oeis.org

1, 1, 0, 1, 5, 63

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}}.

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

Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646, A300913.

cp's OEIS Frontend

A319628 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.

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A319793 Number of non-isomorphic connected strict multiset partitions (sets of multisets) of weight n with empty intersection.

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A319619 Number of non-isomorphic connected weight-n antichains of multisets whose dual is also an antichain of multisets.

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A319620 Number of connected antichain covers of n vertices by distinct sets whose dual is also a (not necessarily strict) antichain.

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A319621 Number of non-isomorphic connected antichain covers of n vertices by distinct sets whose dual is also an antichain of (not necessarily distinct) sets.

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A319622 Number of non-isomorphic connected weight-n antichains of distinct sets whose dual is also an antichain of (not necessarily distinct) sets.

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A319623 Number of connected antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

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A319624 Number of non-isomorphic connected antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

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