cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 21 results. Next

A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 14, 56, 219, 1001, 4588
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 14 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{2}}  {{1}{11}}
                     {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A002846, A005121, A007716, A050343, A213427, A269134, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.
Cf. A323788, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

A323795 Number of non-isomorphic weight-n sets of non-overlapping sets of sets.

Original entry on oeis.org

1, 1, 3, 8, 27, 82, 310, 1163
Offset: 0

Views

Author

Gus Wiseman, Jan 28 2019

Keywords

Comments

Also the number of non-isomorphic set partitions of set-systems of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 27 multiset partitions:
  {{1}}  {{12}}      {{123}}          {{1234}}
         {{1}{2}}    {{1}{12}}        {{1}{123}}
         {{1}}{{2}}  {{1}{23}}        {{12}{13}}
                     {{1}}{{12}}      {{1}{234}}
                     {{1}}{{23}}      {{12}{34}}
                     {{1}{2}{3}}      {{1}}{{123}}
                     {{1}}{{2}{3}}    {{1}{2}{12}}
                     {{1}}{{2}}{{3}}  {{1}{2}{13}}
                                      {{12}}{{13}}
                                      {{1}}{{234}}
                                      {{1}{2}{34}}
                                      {{12}}{{34}}
                                      {{1}}{{2}{12}}
                                      {{12}}{{1}{2}}
                                      {{1}}{{2}{13}}
                                      {{12}}{{1}{3}}
                                      {{1}}{{2}{34}}
                                      {{1}{2}{3}{4}}
                                      {{12}}{{3}{4}}
                                      {{2}}{{1}{13}}
                                      {{1}}{{2}}{{12}}
                                      {{1}}{{2}}{{13}}
                                      {{1}}{{2}}{{34}}
                                      {{1}}{{2}{3}{4}}
                                      {{1}{2}}{{3}{4}}
                                      {{1}}{{2}}{{3}{4}}
                                      {{1}}{{2}}{{3}}{{4}}

Crossrefs

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A318566.
Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323794.

A330474 Number of non-isomorphic balanced reduced multisystems of weight n.

Original entry on oeis.org

1, 1, 2, 7, 48, 424
Offset: 0

Views

Author

Gus Wiseman, Dec 26 2019

Keywords

Comments

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

Examples

			Non-isomorphic representatives of the a(3) = 7 multisystems:
  {1,1,1}
  {1,1,2}
  {1,2,3}
  {{1},{1,1}}
  {{1},{1,2}}
  {{1},{2,3}}
  {{2},{1,1}}
Non-isomorphic representatives of the a(4) = 48 multisystems:
  {1,1,1,1}  {{1},{1,1,1}}    {{{1}},{{1},{1,1}}}
  {1,1,1,2}  {{1,1},{1,1}}    {{{1,1}},{{1},{1}}}
  {1,1,2,2}  {{1},{1,1,2}}    {{{1}},{{1},{1,2}}}
  {1,1,2,3}  {{1,1},{1,2}}    {{{1,1}},{{1},{2}}}
  {1,2,3,4}  {{1},{1,2,2}}    {{{1}},{{1},{2,2}}}
             {{1,1},{2,2}}    {{{1,1}},{{2},{2}}}
             {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
             {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
             {{1,2},{1,2}}    {{{1}},{{2},{1,1}}}
             {{1,2},{1,3}}    {{{1,2}},{{1},{1}}}
             {{1},{2,3,4}}    {{{1}},{{2},{1,2}}}
             {{1,2},{3,4}}    {{{1,2}},{{1},{2}}}
             {{2},{1,1,1}}    {{{1}},{{2},{1,3}}}
             {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
             {{1},{1},{1,1}}  {{{1}},{{2},{3,4}}}
             {{1},{1},{1,2}}  {{{1,2}},{{3},{4}}}
             {{1},{1},{2,2}}  {{{2}},{{1},{1,1}}}
             {{1},{1},{2,3}}  {{{2}},{{1},{1,3}}}
             {{1},{2},{1,1}}  {{{2}},{{3},{1,1}}}
             {{1},{2},{1,2}}  {{{2,3}},{{1},{1}}}
             {{1},{2},{1,3}}
             {{1},{2},{3,4}}
             {{2},{3},{1,1}}

Crossrefs

Labeled versions are A330475 (strongly normal) and A330655 (normal).
The case where the atoms are all different is A318813.
The case where the atoms are all equal is (also) A318813.
The labeled case of set partitions is A005121.
The labeled case of integer partitions is A330679.
The case of maximal depth is A330663.
The version where leaves are sets (as opposed to multisets) is A330668.
Cf. A000311, A000669, A001678, A002846, A004114, A007716, A048816, A213427, A306186, A320154, A320160, A330470, A330666.

A323790 Number of non-isomorphic weight-n sets of sets of sets.

Original entry on oeis.org

1, 1, 3, 9, 33, 113, 474, 1985
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

Non-isomorphic sets of sets are counted by A283877.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 sets of sets of sets:
  {{1}}  {{12}}      {{123}}
         {{1}{2}}    {{1}{12}}
         {{1}}{{2}}  {{1}{23}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}
Non-isomorphic representatives of the a(4) = 33 sets of sets of sets:
  {{1234}}             {{1}{123}}         {{1}{2}{12}}       {{1}}{{1}{12}}
  {{1}{234}}           {{12}{13}}         {{1}}{{2}{12}}
  {{12}{34}}           {{1}}{{123}}       {{12}}{{1}{2}}
  {{1}}{{234}}         {{1}{2}{13}}       {{1}}{{2}}{{12}}
  {{1}{2}{34}}         {{12}}{{13}}       {{1}}{{2}}{{1}{2}}
  {{12}}{{34}}         {{1}}{{1}{23}}
  {{1}}{{2}{34}}       {{1}}{{2}{13}}
  {{1}{2}{3}{4}}       {{12}}{{1}{3}}
  {{12}}{{3}{4}}       {{2}}{{1}{13}}
  {{1}}{{2}}{{34}}     {{1}}{{1}{2}{3}}
  {{1}}{{2}{3}{4}}     {{1}}{{2}}{{13}}
  {{1}{2}}{{3}{4}}     {{1}{2}}{{1}{3}}
  {{1}}{{2}}{{3}{4}}   {{1}}{{2}}{{1}{3}}
  {{1}}{{2}}{{3}}{{4}}

Crossrefs

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A316980, A318564, A318565, A318566, A318812.
Cf. A323787, A323788, A323789, A323791, A323792, A323793, A323794, A323795.

A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.

Original entry on oeis.org

1, 1, 2, 4, 20, 140, 1411
Offset: 0

Views

Author

Gus Wiseman, Dec 27 2019

Keywords

Comments

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

Examples

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 20 multisystems:
  {1,1}  {{1},{1,1}}  {{{1}},{{1},{1,1}}}
  {1,2}  {{1},{1,2}}  {{{1,1}},{{1},{1}}}
         {{1},{2,3}}  {{{1}},{{1},{1,2}}}
         {{2},{1,1}}  {{{1,1}},{{1},{2}}}
                      {{{1}},{{1},{2,2}}}
                      {{{1,1}},{{2},{2}}}
                      {{{1}},{{1},{2,3}}}
                      {{{1,1}},{{2},{3}}}
                      {{{1}},{{2},{1,1}}}
                      {{{1,2}},{{1},{1}}}
                      {{{1}},{{2},{1,2}}}
                      {{{1,2}},{{1},{2}}}
                      {{{1}},{{2},{1,3}}}
                      {{{1,2}},{{1},{3}}}
                      {{{1}},{{2},{3,4}}}
                      {{{1,2}},{{3},{4}}}
                      {{{2}},{{1},{1,1}}}
                      {{{2}},{{1},{1,3}}}
                      {{{2}},{{3},{1,1}}}
                      {{{2,3}},{{1},{1}}}

Crossrefs

The non-maximal version is A330474.
Labeled versions are A330675 (strongly normal) and A330676 (normal).
The case where the leaves are sets (as opposed to multisets) is A330677.
The case with all atoms distinct is A000111.
The case with all atoms equal is (also) A000111.
Cf. A000311, A004114, A005121, A006472, A007716, A048816, A141268, A306186, A330470, A330655, A330664.

A323788 Number of non-isomorphic weight-n sets of multisets of multisets.

Original entry on oeis.org

1, 1, 5, 19, 88, 391, 1995, 10281
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

Also the number of non-isomorphic strict multiset partitions of multiset partitions of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 19 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{1}}    {{123}}
         {{1}{2}}    {{1}{11}}
         {{1}}{{2}}  {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}{1}{1}}
                     {{1}}{{12}}
                     {{1}{1}{2}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{1}{1}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{2}}{{1}{1}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A005121, A007716, A049311, A050343, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.
Cf. A323787, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

A323791 Number of non-isomorphic weight-n sets of multisets of sets.

Original entry on oeis.org

1, 1, 4, 13, 52, 196, 877, 3917
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 13 sets of multisets of sets:
  {{1}}  {{12}}      {{123}}
         {{1}{1}}    {{1}{12}}
         {{1}{2}}    {{1}{23}}
         {{1}}{{2}}  {{1}{1}{1}}
                     {{1}}{{12}}
                     {{1}{1}{2}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{1}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{2}}{{1}{1}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A007716, A049311, A050326, A050343, A283877, A306186, A316980, A318564, A318565, A318566, A318812.
Cf. A323787, A323788, A323789, A323790, A323792, A323793, A323794, A323795.

A323792 Number of non-isomorphic weight-n multisets of sets of sets.

Original entry on oeis.org

1, 1, 4, 11, 43, 145, 614, 2549
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 11 multiset partitions:
  {{1}}  {{12}}      {{123}}
         {{1}{2}}    {{1}{12}}
         {{1}}{{1}}  {{1}{23}}
         {{1}}{{2}}  {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{1}}{{1}}
                     {{1}}{{1}}{{2}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A007716, A049311, A050326, A050343, A255906, A283877, A306186, A316980, A318564, A318565, A318566.
Cf. A323787, A323788, A323789, A323790, A323791, A323793, A323794, A323795.

A323793 Number of non-isomorphic weight-n multisets of multisets of sets.

Original entry on oeis.org

1, 1, 5, 15, 65, 240, 1090, 4845
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

Also the number of non-isomorphic multiset partitions of set multipartitions of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 15 multiset partitions:
  {{1}}  {{12}}      {{123}}
         {{1}{1}}    {{1}{12}}
         {{1}{2}}    {{1}{23}}
         {{1}}{{1}}  {{1}{1}{1}}
         {{1}}{{2}}  {{1}}{{12}}
                     {{1}{1}{2}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{1}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{2}}{{1}{1}}
                     {{1}}{{1}}{{1}}
                     {{1}}{{1}}{{2}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A007716, A049311, A283877, A306186, A316980, A318565, A318566.
Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323794, A323795.

A323794 Number of non-isomorphic weight-n multisets of sets of multisets.

Original entry on oeis.org

1, 1, 5, 17, 77, 318, 1561, 7667
Offset: 0

Views

Author

Gus Wiseman, Jan 28 2019

Keywords

Comments

Also the number of non-isomorphic set multipartitions of multiset partitions of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 17 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{1}}  {{1}{11}}
         {{1}}{{2}}  {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{1}}{{1}}
                     {{1}}{{1}}{{2}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A007716, A049311, A283877, A306186, A316980, A317791, A318565, A318566.
Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323795.
Showing 1-10 of 21 results. Next

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