A054976 -id:A054976 - cp's OEIS frontend

A055084 Number of 6 X n binary matrices with no zero rows or columns, up to row and column permutation.

1, 15, 180, 2001, 20755, 200082, 1781941, 14637962, 111011667, 779695050, 5093379110, 31092553357, 178203364143, 963217652830, 4930357535218, 23989343505296, 111335037107474, 494383391324356, 2106346854756098

Offset: 1

Vladeta Jovovic, Jun 13 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=6 of A056152.

Cf. A024206, A055609, A054976, A048291.

Vec((G(6, x) - G(5, x))*(1 - x) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

A321615 Triangle read by rows: T(n,k) is the number of k X k integer matrices with sum of elements n, 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, 1, 6, 3, 1, 0, 1, 9, 13, 3, 1, 0, 1, 17, 38, 20, 3, 1, 0, 1, 23, 97, 82, 23, 3, 1, 0, 1, 36, 217, 311, 126, 24, 3, 1, 0, 1, 46, 453, 968, 624, 151, 24, 3, 1, 0, 1, 65, 868, 2825, 2637, 933, 162, 24, 3, 1, 0, 1, 80, 1585, 7394, 10098, 4942, 1132, 165, 24, 3, 1

Offset: 0

Andrew Howroyd, Nov 14 2018

Also the number of non-isomorphic multiset partitions of weight n with k parts and k vertices, where the weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Nov 18 2018

			Triangle begins:
    1
    0  1
    0  1    1
    0  1    2    1
    0  1    6    3    1
    0  1    9   13    3    1
    0  1   17   38   20    3    1
    0  1   23   97   82   23    3    1
    0  1   36  217  311  126   24    3    1
    0  1   46  453  968  624  151   24    3    1
    0  1   65  868 2825 2637  933  162   24    3    1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=0..3 are A000007, A000012, A054974, A054975.
Row sums are A319616.
Cf. A007716, A048291, A054976, A057149, A057150, A104601, A120732, A318795, A320808.

(* See A318795 for M[m, n, k]. *)
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, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 24 2018, from PARI *)

\\ See A318795 for M.
T(n, k) = if(k==0, n==0, M(k, k, n) - 2*M(k, k-1, n) + M(k-1, k-1, n));

\\ See A340652 for G.
T(n)={[Vecrev(p) | p<-Vec(1 + sum(k=1, n, y^k*(polcoef(G(k, n, n, y), k, y) - polcoef(G(k-1, n, n, y), k, y))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 16 2024

Column k=0 inserted by Andrew Howroyd, Jan 17 2024

A055082 Number of 4 X n binary matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 8, 42, 179, 633, 2001, 5745, 15274, 38000, 89331, 199715, 427184, 878152, 1741964, 3345562, 6239390, 11327863, 20065972, 34747460, 58924066, 98002370, 160086580, 257148244, 406637336, 633669040, 973971441, 1477810227, 2215179768, 3282598034, 4811946882

Offset: 1

Vladeta Jovovic, Jun 13 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy]

Column k=4 of A056152.

Cf. A024206, A055609, A054976, A048291.

Vec((G(4, x) - G(3, x))*(1 - x) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

Terms a(21) and beyond from Andrew Howroyd, Mar 25 2020

A055083 Number of 5 X n binary matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 11, 91, 633, 3835, 20755, 102089, 461272, 1930310, 7534742, 27602968, 95428291, 312864361, 976985630, 2917175450, 8357692894, 23046527311, 61337188725, 157950527167, 394427897066, 957058104818, 2260601179661, 5206447640059, 11709619965923, 25752660738209

Offset: 1

Vladeta Jovovic, Jun 13 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column k=5 of A056152.

Cf. A024206, A055609, A054976, A048291.

Vec((G(5, x) - G(4, x))*(1 - x) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

Terms a(21) and beyond from Andrew Howroyd, Mar 25 2020

A056037 Number of 6x6 binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 2, 15, 69, 288, 840, 2144, 4488, 8317, 13160, 18636, 23078, 25856, 25623, 23187, 18713, 13932, 9288, 5816, 3256, 1767, 858, 419, 180, 88, 34, 16, 6, 3, 1, 1

Offset: 6

Vladeta Jovovic, Aug 04 2000

Sum_{k=0..36} a(n)=A054976(6).

Cf. A052370.

G.f. : Z(S_6 X S_6; x_1, x_2, ...)-2*Z(S_6 X S_5; x_1, x_2, ...)+Z(S_5 X S_5; x_1, x_2, ...) if we replace x_i by 1+x^i, where Z(S_i X S_j; x_1, x_2, ...) is cycle index of Cartesian product of symmetric groups S_i and S_j of degree i and j, respectively.

A056079 Number of 7 X 7 binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 2, 15, 79, 420, 1744, 6197, 18715, 50042, 118121, 250025, 475791, 820987, 1287695, 1845875, 2423114, 2923474, 3246721, 3327677, 3150758, 2761802, 2242711, 1690526, 1183725, 771951, 469281, 267108, 142542, 71844, 34271, 15685, 6861, 2948, 1223, 513, 209, 90, 34, 16, 6, 3, 1, 1

Offset: 7

Vladeta Jovovic, Aug 04 2000

Sum_{k=0..49} a(n)=A054976(7).

Cf. A053304.

G.f. : Z(S_7 X S_7; x_1, x_2, ...)-2*Z(S_7 X S_6; x_1, x_2, ...)+Z(S_6 X S_6; x_1, x_2, ...) if we replace x_i by 1+x^i, where Z(S_i X S_j; x_1, x_2, ...) is cycle index of Cartesian product of symmetric groups S_i and S_j of degree i and j, respectively.

More terms from Sean A. Irvine, Apr 14 2022

A056080 Number of 5 X 5 binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 2, 14, 49, 131, 248, 410, 531, 601, 566, 474, 336, 222, 124, 67, 32, 16, 6, 3, 1, 1

Offset: 5

Vladeta Jovovic, Aug 04 2000

Sum_{k=0..25} a(n)=A054976(5).

Cf. A052371.

G.f. : Z(S_5 X S_5; x_1, x_2, ...)-2*Z(S_5 X S_4; x_1, x_2, ...)+Z(S_4 X S_4; x_1, x_2, ...) if we replace x_i by 1+x^i, where Z(S_i X S_j; x_1, x_2, ...) is cycle index of Cartesian product of symmetric groups S_i and S_j of degree i and j, respectively.

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

Gus Wiseman, Nov 18 2018

			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]

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

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}]

a(7)-a(14) from Lars Blomberg, May 23 2019

cp's OEIS Frontend

A055084 Number of 6 X n binary matrices with no zero rows or columns, up to row and column permutation.

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A321615 Triangle read by rows: T(n,k) is the number of k X k integer matrices with sum of elements n, with no zero rows or columns, up to row and column permutation.

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A055082 Number of 4 X n binary matrices with no zero rows or columns, up to row and column permutation.

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A055083 Number of 5 X n binary matrices with no zero rows or columns, up to row and column permutation.

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A056037 Number of 6x6 binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

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A056079 Number of 7 X 7 binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

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A056080 Number of 5 X 5 binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

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

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