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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A057151 Number of square binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 41, 102, 252, 666, 1789, 5031, 14486, 43280, 132777, 420267, 1366307, 4566966, 15661086, 55081118, 198425478, 731661754, 2758808581, 10629386376, 41814350148, 167830018952, 686822393793, 2864024856054, 12162059027416, 52564545391789
Offset: 1

Views

Author

Vladeta Jovovic, Aug 14 2000

Keywords

Comments

Number of square binary matrices with n ones and with no zero rows or columns is A104602(n). - Vladeta Jovovic, Mar 25 2006
Also the number of non-isomorphic square set multipartitions (multisets of sets) of weight n. A multiset partition or hypergraph is square if its length (number of blocks or edges) is equal to its number of vertices. The weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Nov 16 2018

Examples

			There are 666 square binary matrices with 10 ones, with no zero rows or columns, up to row and column permutation: 33 of size 4 X 4, 248 of size 5 X 5, 288 of size 6 X 6, 79 of size 7 X 7, 15 of size 8 X 8, 2 of size 9 X 9 and 1 of size 10 X 10. Cf. A057150.
From _Gus Wiseman_, Nov 16 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(6) = 18 square set multipartitions:
  {1}  {1}{2}  {2}{12}    {12}{12}      {1}{23}{23}      {12}{13}{23}
               {1}{2}{3}  {1}{1}{23}    {2}{13}{23}      {1}{23}{123}
                          {1}{3}{23}    {2}{3}{123}      {13}{23}{23}
                          {1}{2}{3}{4}  {3}{13}{23}      {3}{23}{123}
                                        {3}{3}{123}      {1}{1}{1}{234}
                                        {1}{2}{2}{34}    {1}{1}{24}{34}
                                        {1}{2}{4}{34}    {1}{1}{4}{234}
                                        {1}{2}{3}{4}{5}  {1}{2}{34}{34}
                                                         {1}{3}{24}{34}
                                                         {1}{3}{4}{234}
                                                         {1}{4}{24}{34}
                                                         {1}{4}{4}{234}
                                                         {2}{4}{12}{34}
                                                         {3}{4}{12}{34}
                                                         {4}{4}{12}{34}
                                                         {1}{2}{3}{3}{45}
                                                         {1}{2}{3}{5}{45}
                                                         {1}{2}{3}{4}{5}{6}
(End)

Links

Crossrefs

Cf. A049311, A056037, A056079, A056080, A057149, A057150, A057152.
Cf. A054976, A101370, A104601, A104602, A120732, A283877, A319616.

Extensions

More terms from Max Alekseyev, May 31 2007

A057150 Triangle read by rows: T(n,k) = number of k X k binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 5, 2, 1, 0, 0, 4, 11, 2, 1, 0, 0, 3, 21, 14, 2, 1, 0, 0, 1, 34, 49, 15, 2, 1, 0, 0, 1, 33, 131, 69, 15, 2, 1, 0, 0, 0, 33, 248, 288, 79, 15, 2, 1, 0, 0, 0, 19, 410, 840, 420, 82, 15, 2, 1, 0, 0, 0, 14, 531, 2144, 1744, 497, 83, 15, 2, 1
Offset: 1

Views

Author

Vladeta Jovovic, Aug 14 2000

Keywords

Comments

Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts and k vertices. - Gus Wiseman, Nov 14 2018

Examples

			[1], [0,1], [0,1,1], [0,1,2,1], [0,0,5,2,1], [0,0,4,11,2,1], ...;
There are 8 square binary matrices with 5 ones, with no zero rows or columns, up to row and column permutation: 5 of size 3 X 3:
[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
[0 0 1] [0 1 0] [0 1 1] [0 1 1] [1 1 0]
[1 1 1] [1 1 1] [1 0 1] [1 1 0] [1 1 0]
2 of size 4 X 4:
[0 0 0 1] [0 0 0 1]
[0 0 0 1] [0 0 1 0]
[0 0 1 0] [0 1 0 0]
[1 1 0 0] [1 0 0 1]
and 1 of size 5 X 5:
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0].
From _Gus Wiseman_, Nov 14 2018: (Start)
Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   0   5   2   1
   0   0   4  11   2   1
   0   0   3  21  14   2   1
   0   0   1  34  49  15   2   1
   0   0   1  33 131  69  15   2   1
   0   0   0  33 248 288  79  15   2   1
Non-isomorphic representatives of the multiset partitions counted in row 6 {0,0,4,11,2,1} are:
  {{12}{13}{23}}  {{1}{1}{1}{234}}  {{1}{2}{3}{3}{45}}  {{1}{2}{3}{4}{5}{6}}
  {{1}{23}{123}}  {{1}{1}{24}{34}}  {{1}{2}{3}{5}{45}}
  {{13}{23}{23}}  {{1}{1}{4}{234}}
  {{3}{23}{123}}  {{1}{2}{34}{34}}
                  {{1}{3}{24}{34}}
                  {{1}{3}{4}{234}}
                  {{1}{4}{24}{34}}
                  {{1}{4}{4}{234}}
                  {{2}{4}{12}{34}}
                  {{3}{4}{12}{34}}
                  {{4}{4}{12}{34}}
(End)

Crossrefs

Row sums give A057151.
Cf. A049311, A056037, A056079, A056080, A057149, A057151, A057152.
Cf. A007716, A048291, A054976, A101370, A104601, A104602, A120732, A120733, A135588, A319616, A321609, A321615.

Programs

  • Mathematica
    permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
c[p_List, q_List, k_] := SeriesCoefficient[Product[Product[(1 + O[x]^(k + 1) + x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}], {i, 1, Length[p]}], {x, 0, k}];
M[m_, n_, k_] := M[m, n, k] = Module[{s = 0}, Do[Do[s += permcount[p]* permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
T[n_, k_] := M[k, k, n] - 2*M[k, k - 1, n] + M[k - 1, k - 1, n];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)
  • PARI
    \\ See A321609 for M.
T(n,k) = M(k,k,n) - 2*M(k,k-1,n) + M(k-1,k-1,n); \\ Andrew Howroyd, Nov 14 2018

Extensions

Duplicate seventh row removed by Gus Wiseman, Nov 14 2018

A057149 Triangle T(n,k) of n X n binary matrices with k ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 5, 4, 3, 1, 1, 0, 0, 0, 0, 1, 2, 11, 21, 34, 33, 33, 19, 14, 6, 3, 1, 1, 0, 0, 0, 0, 0, 1, 2, 14, 49, 131, 248, 410, 531, 601, 566, 474, 336, 222, 124, 67, 32, 16, 6, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 15, 69, 288, 840, 2144, 4488, 8317, 13160
Offset: 1

Views

Author

Vladeta Jovovic, Aug 14 2000

Keywords

Comments

Row sums give A054976.

Examples

			[0,1], [0,0,1,1,1], [0,0,0,1,2,5,4,3,1,1],...;
T(4,6)=11, i.e. there are 11 4 X 4 binary matrices with 6 ones, with no zero rows or columns, up to row and column permutation:
[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 0 1] [0 0 0 1] [0 0 0 1] [0 0 0 1] [0 0 0 1] [0 0 1 0]
[0 0 0 1] [0 0 1 0] [0 0 1 0] [0 0 1 1] [0 1 1 0] [0 0 1 1]
[1 1 1 0] [1 1 0 1] [1 1 1 0] [1 1 0 0] [1 0 1 0] [1 1 0 0]
and
[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 1 0] [0 0 1 0] [0 0 1 0]
[0 1 0 0] [0 1 0 1] [0 1 0 1] [0 1 0 1] [1 1 0 0]
[1 0 1 1] [1 0 0 1] [1 0 1 0] [1 1 0 0] [1 1 0 0].

Links

Crossrefs

Cf. A056080, A056037, A056079, A057150-A057152, A049311.

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
Showing 1-4 of 4 results.

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