A319781 -id:A319781

A319793

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

+0
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

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

FORMULA

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

EXAMPLE

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}}

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 27 2018

EXTENSIONS

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

STATUS

approved

A319792

Number of non-isomorphic connected set systems of weight n with empty intersection.

+0
1

1, 0, 0, 0, 1, 2, 9, 22, 69, 190, 567, 1640, 5025, 15404, 49048, 159074, 531165, 1813627, 6352739, 22759620, 83443086, 312612543, 1196356133, 4672620842, 18615188819, 75593464871, 312729620542, 1317267618429, 5646454341658, 24618309943464, 109123789229297

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

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

FORMULA

a(n) = A300913(n) - A283877(n) + A319751(n). - Andrew Howroyd, May 31 2023

EXAMPLE

Non-isomorphic representatives of the a(4) = 1 through a(6) = 9 connected set systems:

4: {{1},{2},{1,2}}

5: {{2},{3},{1,2,3}}

{{2},{1,3},{2,3}}

6: {{1},{1,4},{2,3,4}}

{{1},{2,3},{1,2,3}}

{{3},{4},{1,2,3,4}}

{{3},{1,4},{2,3,4}}

{{1,2},{1,3},{2,3}}

{{1,3},{2,4},{3,4}}

{{1},{2},{3},{1,2,3}}

{{1},{2},{1,3},{2,3}}

{{2},{3},{1,3},{2,3}}

CROSSREFS

Cf. A007716, A007718, A049311, A056156, A281116, A283877, A300913, A305854, A316980, A317752, A317755, A317757.

Cf. A319077, A319748, A319751, A319755, A319778, A319781, A319790.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 27 2018

EXTENSIONS

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

STATUS

approved

A319791

Number of non-isomorphic connected set multipartitions (multisets of sets) of weight n with empty intersection.

+0
5

1, 0, 0, 0, 1, 3, 14, 38, 125, 360, 1107, 3297, 10292, 32134, 103759, 340566, 1148150, 3951339, 13925330, 50122316, 184365292, 692145409, 2651444318, 10356184440, 41224744182, 167150406897, 689998967755, 2898493498253, 12384852601731, 53804601888559, 237566072006014

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

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

FORMULA

a(n) = A056156(n) - A049311(n) + A319748(n). - Andrew Howroyd, May 31 2023

EXAMPLE

Non-isomorphic representatives of the a(4) = 1 through a(6) = 14 set multipartitions:

4: {{1},{2},{1,2}}

5: {{2},{3},{1,2,3}}

{{2},{1,3},{2,3}}

{{1},{2},{2},{1,2}}

6: {{1},{1,4},{2,3,4}}

{{1},{2,3},{1,2,3}}

{{3},{4},{1,2,3,4}}

{{3},{1,4},{2,3,4}}

{{1,2},{1,3},{2,3}}

{{1,3},{2,4},{3,4}}

{{1},{2},{3},{1,2,3}}

{{1},{2},{1,2},{1,2}}

{{1},{2},{1,3},{2,3}}

{{2},{2},{1,3},{2,3}}

{{2},{3},{3},{1,2,3}}

{{2},{3},{1,3},{2,3}}

{{1},{1},{2},{2},{1,2}}

{{1},{2},{2},{2},{1,2}}

CROSSREFS

Cf. A007716, A007718, A049311, A056156, A281116, A283877, A317752, A317755, A317757.

Cf. A319077, A319748, A319755, A319778, A319781, A319790.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 27 2018

EXTENSIONS

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

STATUS

approved

A319790

Number of non-isomorphic connected multiset partitions of weight n with empty intersection.

+0
5

1, 0, 0, 0, 1, 5, 32, 134, 588, 2335, 9335, 36506, 144263, 571238, 2291894, 9300462, 38303796, 160062325, 679333926, 2927951665, 12817221628, 56974693933, 257132512297, 1177882648846, 5475237760563, 25818721638720, 123473772356785, 598687942799298, 2942344764127039

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

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

FORMULA

a(n) = A007718(n) - A007716(n) + A317757(n). - Andrew Howroyd, May 31 2023

EXAMPLE

Non-isomorphic representatives of the a(4) = 1 through a(5) = 5 connected 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}}

{{1},{2},{2},{1,2}}

CROSSREFS

Cf. A007716, A007718, A049311, A056156, A283877, A317752, A317755, A317757.

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 27 2018

EXTENSIONS

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

STATUS

approved

A319783

Number of set systems spanning n vertices with empty intersection whose dual is also a set system with empty intersection.

+0
4

1, 0, 0, 1, 203, 490572

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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}}.

LINKS

Table of n, a(n) for n=0..5.

EXAMPLE

The a(3) = 1 set system is {{1,2},{1,3},{2,3}}.

CROSSREFS

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757.

Cf. A319775, A319778, A319779, A319781.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319779

Number of intersecting multiset partitions of weight n whose dual is not an intersecting multiset partition.

+0
16

1, 0, 0, 0, 1, 4, 20, 66, 226, 696, 2156

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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.

A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

LINKS

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

EXAMPLE

Non-isomorphic representatives of the a(4) = 1 through a(6) = 20 multiset partitions:

4: {{1,3},{2,3}}

5: {{1,2},{2,3,3}}

{{1,3},{2,3,3}}

{{1,4},{2,3,4}}

{{3},{1,3},{2,3}}

6: {{1,2},{2,3,3,3}}

{{1,3},{2,2,3,3}}

{{1,3},{2,3,3,3}}

{{1,3},{2,3,4,4}}

{{1,4},{2,3,4,4}}

{{1,5},{2,3,4,5}}

{{1,1,2},{2,3,3}}

{{1,2,2},{2,3,3}}

{{1,2,3},{3,4,4}}

{{1,2,4},{3,4,4}}

{{1,2,5},{3,4,5}}

{{1,3,3},{2,3,3}}

{{1,3,4},{2,3,4}}

{{2},{1,2},{2,3,3}}

{{3},{1,3},{2,3,3}}

{{4},{1,4},{2,3,4}}

{{1,3},{2,3},{2,3}}

{{1,3},{2,3},{3,3}}

{{1,4},{2,4},{3,4}}

{{3},{3},{1,3},{2,3}}

CROSSREFS

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319775, A319778, A319781, A319783.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319778

Number of non-isomorphic set systems of weight n with empty intersection whose dual is also a set system with empty intersection.

+0
12

1, 0, 1, 1, 2, 5, 13, 28, 72, 181, 483

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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 dual of a multiset partition has empty intersection iff no part contains all the 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.

LINKS

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

EXAMPLE

Non-isomorphic representatives of the a(2) = 1 through a(6) = 13 multiset partitions:

2: {{1},{2}}

3: {{1},{2},{3}}

4: {{1},{3},{2,3}}

{{1},{2},{3},{4}}

5: {{1},{2,4},{3,4}}

{{2},{1,3},{2,3}}

{{1},{2},{3},{2,3}}

{{1},{2},{4},{3,4}}

{{1},{2},{3},{4},{5}}

6: {{3},{1,4},{2,3,4}}

{{1,2},{1,3},{2,3}}

{{1,3},{2,4},{3,4}}

{{1},{2},{1,3},{2,3}}

{{1},{2},{3,5},{4,5}}

{{1},{3},{4},{2,3,4}}

{{1},{3},{2,4},{3,4}}

{{1},{4},{2,4},{3,4}}

{{2},{3},{1,3},{2,3}}

{{2},{4},{1,2},{3,4}}

{{1},{2},{3},{4},{3,4}}

{{1},{2},{3},{5},{4,5}}

{{1},{2},{3},{4},{5},{6}}

CROSSREFS

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757.

Cf. A319775, A319779, A319781, A319783.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319775

Number of non-isomorphic multiset partitions of weight n with empty intersection and no part containing all the vertices.

+0
4

1, 0, 1, 4, 16, 52, 185, 625, 2226, 7840, 28405

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

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

EXAMPLE

Non-isomorphic representatives of the a(2) = 1 through a(4) = 16 multiset partitions:

2: {{1},{2}}

3: {{1},{2,2}}

{{1},{2,3}}

{{1},{2},{2}}

{{1},{2},{3}}

4: {{1},{2,2,2}}

{{1},{2,3,3}}

{{1},{2,3,4}}

{{1,1},{2,2}}

{{1,2},{3,3}}

{{1,2},{3,4}}

{{1},{1},{2,2}}

{{1},{1},{2,3}}

{{1},{2},{2,2}}

{{1},{2},{3,3}}

{{1},{2},{3,4}}

{{1},{3},{2,3}}

{{1},{1},{2},{2}}

{{1},{2},{2},{2}}

{{1},{2},{3},{3}}

{{1},{2},{3},{4}}

CROSSREFS

Cf. A007716, A281116, A283877, A316980, A317752, A317755, A317757.

Cf. A319778, A319779, A319781, A319783.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319764

Number of non-isomorphic intersecting set systems of weight n with empty intersection.

+0
4

1, 0, 0, 0, 0, 0, 1, 1, 3, 8, 18

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

COMMENTS

A set system is a finite set of finite nonempty sets. It is intersecting if no two parts are disjoint. The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

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

EXAMPLE

Non-isomorphic representatives of the a(6) = 1 through a(9) = 8 set systems:

6: {{1,2},{1,3},{2,3}}

7: {{1,3},{1,4},{2,3,4}}

8: {{1,2},{1,3,4},{2,3,4}}

{{1,4},{1,5},{2,3,4,5}}

{{2,4},{1,2,5},{3,4,5}}

9: {{1,3},{1,4,5},{2,3,4,5}}

{{1,5},{1,6},{2,3,4,5,6}}

{{2,5},{1,2,6},{3,4,5,6}}

{{1,2,3},{2,4,5},{3,4,5}}

{{1,3,5},{2,3,6},{4,5,6}}

{{1,2},{1,3},{1,4},{2,3,4}}

{{1,2},{1,3},{2,3},{1,2,3}}

{{1,3},{1,4},{3,4},{2,3,4}}

CROSSREFS

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317752, A317755, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319763, A319765, A319779, A319781.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319763

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

+0
4

1, 0, 0, 0, 0, 0, 1, 2, 12, 46, 181

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

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

EXAMPLE

Non-isomorphic representatives of the a(6) = 1 through a(8) = 12 multiset partitions:

6: {{1,2},{1,3},{2,3}}

7: {{1,2},{1,3},{2,3,3}}

{{1,3},{1,4},{2,3,4}}

8: {{1,2},{1,3},{2,2,3,3}}

{{1,2},{1,3},{2,3,3,3}}

{{1,2},{1,3},{2,3,4,4}}

{{1,2},{1,3,3},{2,3,3}}

{{1,2},{1,3,4},{2,3,4}}

{{1,3},{1,4},{2,3,4,4}}

{{1,3},{1,1,2},{2,3,3}}

{{1,3},{1,2,2},{2,3,3}}

{{1,4},{1,5},{2,3,4,5}}

{{2,3},{1,2,4},{3,4,4}}

{{2,4},{1,2,3},{3,4,4}}

{{2,4},{1,2,5},{3,4,5}}

CROSSREFS

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319764, A319765, A319779, A319781.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved