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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A321655 Number of distinct row/column permutations of strict plane partitions of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 33, 53, 77, 225
Offset: 0

Views

Author

Gus Wiseman, Nov 15 2018

Keywords

Examples

			The a(6) = 9 permutations of strict plane partitions:
  [6] [2 4] [4 2] [1 5] [5 1] [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
.
  [1] [5] [0 1] [1 0] [2 3] [3 2] [2] [4] [0 2] [1 3] [2 0] [3 1]
  [5] [1] [2 3] [3 2] [0 1] [1 0] [4] [2] [1 3] [0 2] [3 1] [2 0]
.
  [1] [1] [2] [2] [3] [3]
  [2] [3] [1] [3] [1] [2]
  [3] [2] [3] [1] [2] [1]

Crossrefs

Cf. A000219, A001970, A007716, A008480, A068313, A114736, A117433, A120733, A321645, A319646, A321647, A321648.

Programs

  • Mathematica
    submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@Length/@Split[#],OrderedQ[Sort[Map[Last,GatherBy[Sort[Reverse/@#],First],{2}],submultisetQ],submultisetQ],OrderedQ[Sort[Sort/@Map[Last,GatherBy[#,First],{2}],submultisetQ],submultisetQ]]&]],{n,5}]

A321652 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

Original entry on oeis.org

1, 1, 5, 19, 107, 573, 4050, 29093, 249301, 2271020, 23378901, 257871081, 3132494380, 40693204728, 572089068459, 8566311524788, 137165829681775, 2327192535461323, 41865158805428687, 793982154675640340, 15863206077534914434, 332606431999260837036, 7309310804287502958322, 167896287022455809865568
Offset: 0

Views

Author

Gus Wiseman, Nov 15 2018

Keywords

Examples

			The a(3) = 19 matrices:
  [3] [2 1] [1 1 1]
.
  [2] [2 0] [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Links

Crossrefs

Cf. A000219, A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321653, A321654, A321655.

Programs

  • Mathematica
    prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]]]&]],{n,6}]

Formula

Sum of coefficients in the expansions of all homogeneous symmetric functions in terms of monomial symmetric functions. In other words, if Sum_{|y| = n} h(y) = Sum_{|y| = n} c_y * m(y), then a(n) = Sum_{|y| = n} c_y.

Extensions

a(10) onwards from Ludovic Schwob, Aug 29 2023

A321654 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with distinct row sums and distinct column sums.

Original entry on oeis.org

1, 1, 1, 13, 13, 45, 681, 885, 2805, 8301, 237213
Offset: 0

Views

Author

Gus Wiseman, Nov 15 2018

Keywords

Examples

			The a(3) = 13 matrices:
  [3] [2 1] [1 2]
.
  [2] [2 0] [1 1] [1 1] [1] [1 0] [1 0] [0 2] [0 1] [0 1]
  [1] [0 1] [1 0] [0 1] [2] [1 1] [0 2] [1 0] [2 0] [1 1]

Crossrefs

Cf. A000219, A001970, A007716, A068313, A114736, A117433, A120733, A319646, A321645, A321646, A321652.

Programs

  • Mathematica
    prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@Total/@prs2mat[#],UnsameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

A319629 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 9, 29, 66, 189
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(7) = 9 antichains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
4: {{1,1,1,1}}
5: {{1,1,1,1,1}}
   {{1,1},{1,2,2}}
6: {{1,1,1,1,1,1}}
   {{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}}
7: {{1,1,1,1,1,1,1}}
   {{1,1},{1,2,2,2,2}}
   {{1,1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2,2}}
   {{1,1,2},{2,2,2,2}}
   {{1,1,2},{2,3,3,3}}
   {{1,1},{1,2},{2,2,2}}
   {{1,1},{1,2},{2,3,3}}
   {{1,2},{1,3},{2,3,3}}

Crossrefs

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

Formula

Euler transform is A319644.

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

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(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}}

Crossrefs

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

Formula

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

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(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}}

Crossrefs

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

Formula

Euler transform is A319641.

A319638 Number of non-isomorphic weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.

Original entry on oeis.org

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

Crossrefs

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

Formula

Euler transform of A319625.

A319641 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, 3, 5, 11, 18, 41, 70, 159, 323, 778
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(4) = 11 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,3}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,3,4}}
   {{1},{2,2,2}}
   {{1},{2,3,4}}
   {{1,1},{2,2}}
   {{1,2},{3,3}}
   {{1,2},{3,4}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{2},{3},{4}}

Crossrefs

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

Formula

Euler transform of A319628.

A319643 Number of non-isomorphic weight-n strict multiset partitions whose dual is an antichain of (not necessarily distinct) multisets.

Original entry on oeis.org

1, 1, 3, 6, 15, 29, 82, 179, 504, 1302, 3822
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.
From Gus Wiseman, Aug 15 2019: (Start)
Also the number of non-isomorphic T_0 weak antichains of weight n. The T_0 condition means that the dual is strict (no repeated edges). A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other. For example, non-isomorphic representatives of the a(0) = 1 through a(4) = 15 T_0 weak antichains are:
{} {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1},{1}} {{1,2,2}} {{1,2,2,2}}
{{1},{2}} {{1},{2,2}} {{1,1},{1,1}}
{{1},{1},{1}} {{1,1},{2,2}}
{{1},{2},{2}} {{1},{2,2,2}}
{{1},{2},{3}} {{1,2},{2,2}}
{{1},{2,3,3}}
{{1,3},{2,3}}
{{1},{1},{2,2}}
{{1},{2},{3,3}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
(End)

Examples

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

Crossrefs

Cf. A006126, A007716, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646.
Cf. A245567, A293993, A319721, A326704, A326950, A326973, A326978.

A319644 Number of non-isomorphic weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 18, 31, 73, 162, 413
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) = 8 antichains:
1: {{1}}
2: {{1,1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1},{2,2}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1},{2,2,2}}
   {{1,1},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3},{4}}
5: {{1,1,1,1,1}}
   {{1},{2,2,2,2}}
   {{1,1},{1,2,2}}
   {{1,1},{2,2,2}}
   {{1},{2},{3,3,3}}
   {{1},{2,2},{3,3}}
   {{1},{2},{3},{4,4}}
   {{1},{2},{3},{4},{5}}

Crossrefs

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

Formula

Euler transform of A319629.
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