A008300 -id:A008300 - cp's OEIS frontend

A172538 Number of n X n 0..1 arrays with row sums 13 and column sums 13.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 87178291200, 147320988741542099484000, 190637228506535883540302038364160000, 238871596129285108315684789088803525762942560000

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..19

Column k=13 of A008300.

A172540 Number of n X n 0..1 arrays with row sums 9 and column sums 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 3628800, 158815387962000, 9254768770160124288000, 769237071909157579108571190000, 96986285294151066094112970262797953280, 19092174983817380047229162651397270697765056000

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..20

Column k=9 of A008300.

A283627 The number of (n^2) X (n^2) real {0,1}-matrices the square of which is the all-ones matrix.

Original entry on oeis.org

1, 12, 1330560

Offset: 1

R. J. Mathar, Mar 12 2017

These are real {0,1} matrices A such that A^2 = J, the all-ones matrix.
It is known that A must have dimension which is a square, so the sequence shows counts for 1 X 1, 4 X 4, 9 X 9, 16 X 16 matrices and so on.
It is also known that if A has dimension n^2 X n^2, then each row and column must contain exactly n 1's, and A has trace n.
a(3) = 1330560 is confirmed by W. Edwin Clark, Mar 12 2017, who says: (Start)
My method was to take the 6 matrices A1, A2, A3, A4, A5, A6 found by Knuth, which are representatives for the 6 distinct orbits of 9 x 9 matrices A such that A^2 = J under the action of the 9! permutation matrices acting by conjugation.
I found for each Ai the size n_i of the stabilizer of Ai. The stabilizer orders are [n_1,n_2,n_3,n_4,n_5,n_6] = [6,2,1,1,2,2], which implies that the cardinality of the union of all orbits is Sum(9!/n_i, i=1..6) = 1330560.
(End)

			Four of the 12 solutions for 4 X 4 are
  1 1 0 0
  0 0 1 1
  1 1 0 0
  0 0 1 1
.
  1 1 0 0
  0 0 1 1
  0 0 1 1
  1 1 0 0
.
  1 0 0 1
  0 1 1 0
  1 0 0 1
  0 1 1 0
.
  0 1 0 1
  1 0 1 0
  1 0 1 0
  0 1 0 1
.
Solutions for 9 X 9 are, for example,
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  1 1 1 0 0 0 0 0 0
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
.
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  1 1 1 0 0 0 0 0 0
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  0 0 0 1 0 0 0 1 1
  0 0 0 0 1 1 1 0 0
.
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  1 1 1 0 0 0 0 0 0
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 0 0 1
  0 0 0 0 1 0 1 1 0
  0 0 0 1 0 1 0 0 1
  0 0 0 0 1 0 1 1 0

F. Curtis, J. Drew, Central groupoids, central digraphs, and zero-one matrices A satisfying A^2=J, (2002).
F. Curtis, J. Drew, C-K Li, D. Pragel, Central groupoids, central digraphs, and zero-one matrices A satisfying A^2=J, J. Combin. Theo. A (105) (2004) 35-50.
J. Knuth, Notes on Central Groupoids, J. Combin. Theo. 8 (1970) 376-390.
Donald E. Knuth and Peter B. Bendix, Simple word problems in universal algebras, in John Leech (ed.), Computational Problems in Abstract Algebra, Pergamon, 1970, pp. 263-297.
H. Ryser, A generalization of the matrix equation A^2=J, Lin. Algebra Applic. 3 (4) (1970) 451-460.
Y.-K. Wu, R.-Z. Jia, Q. Li, g-circulant solutions to the (0,1) matrix equation A^m=J_n, Lin. Alg. Applic. 345 (1-3) (2002) 195-224.

Cf. A008300.

A364068 Triangle T(n,k) read by rows: Number of traceless binary n X n matrices with all row and column sums equal to k, 1<=k<=n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 9, 9, 1, 0, 44, 216, 44, 1, 0, 265, 7570, 7570, 265, 1, 0, 1854, 357435, 1975560, 357435, 1854, 1, 0, 14833, 22040361, 749649145, 749649145, 22040361, 14833, 1, 0, 133496, 1721632024

Offset: 1

R. J. Mathar, Jul 04 2023

			    0
    1        0
    2        1         0
    9        9         1      0
   44      216        44      1    0
  265     7570      7570    265    1 0
 1854   357435   1975560 357435 1854 1 0
14833 22040361 749649145

Cf. A000166 (k=1), A007107 (k=2), A284989 (see 1st col), A284990 (see 1st col, k=3), A007105 (k=3?), A284991 (see 1st col, k=4), A008300 (any trace)

T(n,n)=0. (k=n would require a 1 on the diagonal)
T(n,n-1)=1. (1 at all entries but the diagonal)
T(n,n-k) = T(n,k-1). (Flip entries 0<->1 and erase diagonal) - R. J. Mathar, Jul 26 2023

A172539 Number of n X n 0..1 arrays with row sums 10 and column sums 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 39916800, 21959547410077200, 14255616537578735986867200, 12163525741347497524178307740904300, 14962628816774970940772777740084998521738256

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..19

Column k=10 of A008300.

A172542 Number of n X n 0..1 arrays with row sums 16 and column sums 16.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 355687428096000, 3237415247416050491577971184000, 13564406360915457771720399143711430952267776000, 37911589613425952733393718264069147678877877626169022024515000

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Brendan D. McKay, Table of n, a(n) for n = 0..30 (terms n=1..20 from R. H. Hardin)
E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 0-1 matrices with equal row and column sums
Shalosh B. Ekhad and Doron Zeilberger, In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses, Maple package Bipartite.
B. D. McKay, 0-1 matrices with constant row and column sums

Column k=16 of A008300.

A172543 Number of n X n 0..1 arrays with row sums 15 and column sums 15.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20922789888000, 10268902998771351157327104000, 2767806480542211571651550187279222472704000

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..20

Column k=15 of A008300.

A383256 Number of n X n matrices of nonnegative entries with all columns summing to n and no horizontally adjacent zeros.

Original entry on oeis.org

1, 1, 7, 343, 125465, 366908001, 8698468668251, 1708834003295306868, 2810884261025802145414705, 39088555382409783097546399456477, 4626844513673581956954679383115038810744, 4688191496359773864437279635019555242588548880831

Offset: 0

John Tyler Rascoe, Apr 21 2025

nonn

			a(1) = 1: [1]
a(2) = 7: [1,1]   [1,0]   [1,2]   [0,1]   [2,1]   [0,2]   [2,0]
          [1,1],  [1,2],  [1,0],  [2,1],  [0,1],  [2,0],  [0,2].

John Tyler Rascoe, Python program.

Cf. A008300, A120733, A145839, A261780, A382923.

Python
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a(10)-a(11) from Bert Dobbelaere, Apr 23 2025

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A172538 Number of n X n 0..1 arrays with row sums 13 and column sums 13.

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A172540 Number of n X n 0..1 arrays with row sums 9 and column sums 9.

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A283627 The number of (n^2) X (n^2) real {0,1}-matrices the square of which is the all-ones matrix.

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A364068 Triangle T(n,k) read by rows: Number of traceless binary n X n matrices with all row and column sums equal to k, 1<=k<=n.

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A172539 Number of n X n 0..1 arrays with row sums 10 and column sums 10.

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A172542 Number of n X n 0..1 arrays with row sums 16 and column sums 16.

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A172543 Number of n X n 0..1 arrays with row sums 15 and column sums 15.

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A383256 Number of n X n matrices of nonnegative entries with all columns summing to n and no horizontally adjacent zeros.

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