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.

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A319634 Number of non-isomorphic 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, 2, 4, 12, 87
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2018

Keywords

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}}.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 antichain covers:
  {{1}}   {{1,2}}     {{1,2,3}}              {{1,2,3,4}}
         {{1},{2}}   {{1},{2,3}}            {{1},{2,3,4}}
                    {{1},{2},{3}}           {{1,2},{3,4}}
                 {{1,2},{1,3},{2,3}}       {{1},{2},{3,4}}
                                          {{1},{2},{3},{4}}
                                       {{1,2},{1,3,4},{2,3,4}}
                                       {{1},{2,3},{2,4},{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}}

Crossrefs

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

A319635 Number of non-isomorphic 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, 3, 5, 7, 12, 16, 26, 36, 58
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2018

Keywords

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.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 7 antichains:
1: {{1}}
2: {{1,2}}
   {{1},{2}}
3: {{1,2,3}}
   {{1},{2,3}}
   {{1},{2},{3}}
4: {{1,2,3,4}}
   {{1},{2,3,4}}
   {{1,2},{3,4}}
   {{1},{2},{3,4}}
   {{1},{2},{3},{4}}
5: {{1,2,3,4,5}}
   {{1},{2,3,4,5}}
   {{1,2},{3,4,5}}
   {{1},{2},{3,4,5}}
   {{1},{2,3},{4,5}}
   {{1},{2},{3},{4,5}}
   {{1},{2},{3},{4},{5}}

Crossrefs

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

A319640 Number of non-isomorphic antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

Original entry on oeis.org

1, 1, 1, 2, 7, 70
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2018

Keywords

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}}.

Examples

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

Crossrefs

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

A319642 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is a chain of (not necessarily distinct) multisets.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 25, 42, 66, 108
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2018

Keywords

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.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1,2},{2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1,2,2,3,3}}
   {{1,2,3,3,3}}
   {{1,2,3,4,4}}
   {{1,2,3,4,5}}
   {{1,2},{2,2,2}}
   {{3,3},{1,2,3}}

Crossrefs

Cf. A000219, A006126, A007716, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646.

A319645 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is a chain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 9, 16, 22, 38
Offset: 0

Views

Author

Gus Wiseman, Sep 25 2018

Keywords

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. 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.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 7 antichains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1,2},{2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1,2},{2,2,2}}
6: {{1,1,1,1,1,1}}
   {{1,1,2,2,2,2}}
   {{1,2,2,2,2,2}}
   {{1,2,2,3,3,3}}
   {{1,2},{2,2,2,2}}
   {{1,2,2},{2,2,2}}
   {{1,2,3},{2,3,3}}

Crossrefs

Cf. A000219, A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319616-A319646.

A323586 Number of plane partitions of n with no repeated rows (or, equivalently, no repeated columns).

Original entry on oeis.org

1, 1, 2, 5, 8, 16, 30, 53, 89, 158, 265, 443, 735, 1197
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2019

Keywords

Examples

			The a(4) = 8 plane partitions with no repeated rows:
  4   31   22   211   1111
.
  3   21   111
  1   1    1
The a(6) = 30 plane partitions with no repeated columns:
  6   51   42   321
.
  5   4   41   3   31   32   31   22   21   221   211
  1   2   1    3   2    1    11   2    21   1     11
.
  4   3   31   2   21   22   21   111
  1   2   1    2   2    1    11   11
  1   1   1    2   1    1    1    1
.
  3   2   21   11
  1   2   1    11
  1   1   1    1
  1   1   1    1
.
  2   11
  1   1
  1   1
  1   1
  1   1
.
  1
  1
  1
  1
  1
  1

Crossrefs

Cf. A000219, A003293 (strict rows), A114736 (strict rows and columns), A117433 (distinct entries), A299968, A319646 (no repeated rows or columns), A323429, A323436 (plane partitions of type), A323580, A323587.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[UnsameQ@@#,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}]
Previous Showing 31-36 of 36 results.

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