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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A321735 Number of (0,1)-matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.

Original entry on oeis.org

1, 1, 2, 7, 30, 153, 939, 6653, 53743, 486576
Offset: 0

Views

Author

Gus Wiseman, Nov 18 2018

Keywords

Examples

			The a(3) = 7 matrices:
  [1 1]
  [1 0]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Crossrefs

Cf. A000700, A007016, A049311, A054976, A057151, A104602, A320451, A321719, A321723, A321732, A321733, A321736, A321739.

Programs

  • Mathematica
    prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

Formula

Let c(y) be the coefficient of m(y) in e(y), where m is monomial symmetric functions and e is elementary symmetric functions. Then a(n) = Sum_{|y| = n} c(y).

A321739 Number of non-isomorphic weight-n set multipartitions (multisets of sets) whose part-sizes are also their vertex-degrees.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 12, 21, 46, 94, 208
Offset: 0

Views

Author

Gus Wiseman, Nov 19 2018

Keywords

Comments

Also the number of (0,1) square matrices up to row and column permutations with n ones and no zero rows or columns, with the same multiset of row sums as of column sums.
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) = 12 set multipartitions:
  {1}  {1}{2}  {2}{12}    {12}{12}      {1}{23}{23}      {12}{13}{23}
               {1}{2}{3}  {1}{1}{23}    {2}{13}{23}      {3}{23}{123}
                          {1}{3}{23}    {3}{3}{123}      {1}{1}{1}{234}
                          {1}{2}{3}{4}  {1}{2}{2}{34}    {1}{1}{24}{34}
                                        {1}{2}{4}{34}    {1}{2}{34}{34}
                                        {1}{2}{3}{4}{5}  {1}{3}{24}{34}
                                                         {1}{4}{4}{234}
                                                         {2}{4}{12}{34}
                                                         {3}{4}{12}{34}
                                                         {1}{2}{3}{3}{45}
                                                         {1}{2}{3}{5}{45}
                                                         {1}{2}{3}{4}{5}{6}

Crossrefs

Cf. A000700, A049311, A057151, A104602, A319056, A320451, A321719, A321721, A321723, A321732, A321734, A321735, A321736, A321854.

A057152 Limiting number of m X m binary matrices with m+n ones, with no zero rows or columns, up to row and column permutations, as m tends to infinity.

Original entry on oeis.org

1, 2, 15, 83, 545, 3493, 24006, 169419, 1249225, 9542846, 75621458, 620011391, 5253319121
Offset: 0

Views

Author

Vladeta Jovovic, Aug 14 2000

Keywords

Crossrefs

Cf. A056080, A056037, A056079, A057149-A057151, A049311.

Formula

a(n) = limit of A057149(m,m+n) as m tends to infinity
a(n) = A057149(3*n,4*n)

Extensions

Formula and a(6)-a(8) added by Max Alekseyev, Jan 24 2010
a(9)-a(12) from Max Alekseyev, Feb 17 2010

A321733 Number of (0,1)-matrices with n ones, no zero rows or columns, and the same row sums as column sums.

Original entry on oeis.org

1, 1, 2, 8, 40, 246, 1816, 15630, 153592, 1696760, 20816358, 280807868, 4131117440, 65823490088, 1129256780408
Offset: 0

Views

Author

Gus Wiseman, Nov 18 2018

Keywords

Examples

			The a(4) = 40 matrices:
  [1 1]
  [1 1]
.
  [1 1 0][1 1 0][1 0 1][1 0 1][1 0 0]
  [1 0 0][0 0 1][1 0 0][0 1 0][0 1 1]
  [0 0 1][1 0 0][0 1 0][1 0 0][0 1 0]
.
  [1 0 0][0 1 1][0 1 0][0 1 0][0 1 0]
  [0 0 1][1 0 0][1 1 0][1 0 1][0 1 1]
  [0 1 1][1 0 0][0 0 1][0 1 0][1 0 0]
.
  [0 1 0][0 0 1][0 0 1][0 0 1][0 0 1]
  [0 0 1][1 1 0][1 0 0][0 1 0][0 0 1]
  [1 0 1][0 1 0][0 1 1][1 0 1][1 1 0]
.
  [1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0]
  [0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]
  [0 0 1 0][0 0 0 1][0 1 0 0][0 0 0 1][0 1 0 0][0 0 1 0]
  [0 0 0 1][0 0 1 0][0 0 0 1][0 1 0 0][0 0 1 0][0 1 0 0]
.
  [0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0]
  [1 0 0 0][1 0 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]
  [0 0 1 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 0 1 0]
  [0 0 0 1][0 0 1 0][0 0 0 1][1 0 0 0][0 0 1 0][1 0 0 0]
.
  [0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0]
  [1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 0 1][0 0 0 1]
  [0 1 0 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 1 0 0]
  [0 0 0 1][0 1 0 0][0 0 0 1][1 0 0 0][0 1 0 0][1 0 0 0]
.
  [0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1]
  [1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0]
  [0 1 0 0][0 0 1 0][1 0 0 0][0 0 1 0][1 0 0 0][0 1 0 0]
  [0 0 1 0][0 1 0 0][0 0 1 0][1 0 0 0][0 1 0 0][1 0 0 0]

Crossrefs

Cf. A006052, A007016, A049311, A054976, A057151, A104602, A120732, A319056, A321717, A321723, A321732, A321735, A321736, A321739.

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[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

Extensions

a(7)-a(14) from Lars Blomberg, May 23 2019
Previous Showing 11-14 of 14 results.

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