A319565 -id:A319565 - cp's OEIS frontend

A321681 Number of non-isomorphic weight-n connected strict antichains of multisets with multiset density -1.

1, 1, 2, 3, 7, 13, 35, 77, 205, 517, 1399

Offset: 0

Gus Wiseman, Nov 16 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

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) = 13 trees:
  {{1}}  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
         {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
                  {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                             {{1,2,3,3}}    {{1,2,2,3,3}}
                             {{1,2,3,4}}    {{1,2,3,3,3}}
                             {{1,2},{2,2}}  {{1,2,3,4,4}}
                             {{1,3},{2,3}}  {{1,2,3,4,5}}
                                            {{1,1},{1,2,2}}
                                            {{1,2},{2,2,2}}
                                            {{1,2},{2,3,3}}
                                            {{1,3},{2,3,3}}
                                            {{1,4},{2,3,4}}
                                            {{3,3},{1,2,3}}

Cf. A006126, A007718, A056156, A096827, A285572, A293993, A293994, A305052, A319557, A319565, A319719, A319721, A321585, A321680.

A319625 Number of non-isomorphic connected weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 3

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) = 3 antichains:
               {{1}}
        {{1,2},{1,3},{2,3}}
     {{1,2},{1,3},{2,4},{3,4}}
    {{1,2},{1,3},{1,4},{2,3,4}}
   {{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, A319558, A319565, A319616-A319646.

Euler transform is A319638.

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

Original entry on oeis.org

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.

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.

A321484 Number of non-isomorphic self-dual connected multiset partitions of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 20, 35, 78, 141

Offset: 0

Gus Wiseman, Nov 16 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts 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) = 9 multiset partitions:
  {{1}}  {{11}}  {{111}}    {{1111}}    {{11111}}      {{111111}}
                 {{2}{12}}  {{12}{12}}  {{11}{122}}    {{112}{122}}
                            {{2}{122}}  {{12}{122}}    {{12}{1222}}
                                        {{2}{1222}}    {{2}{12222}}
                                        {{2}{13}{23}}  {{22}{1122}}
                                        {{3}{3}{123}}  {{12}{13}{23}}
                                                       {{2}{13}{233}}
                                                       {{3}{23}{123}}
                                                       {{3}{3}{1233}}

Cf. A007718, A056156, A138178, A316983, A319565, A319616, A319647, A319719, A321194, A321585, A321680, A321681.

cp's OEIS Frontend

A321681 Number of non-isomorphic weight-n connected strict antichains of multisets with multiset density -1.

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A319625 Number of non-isomorphic connected weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.

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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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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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A321484 Number of non-isomorphic self-dual connected multiset partitions of weight n.

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