A058527 -id:A058527 - cp's OEIS frontend

A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 24, 90, 24, 1, 1, 120, 2040, 2040, 120, 1, 1, 720, 67950, 297200, 67950, 720, 1, 1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1, 1, 40320, 187530840, 24046189440, 116963796250, 24046189440, 187530840, 40320, 1, 1, 362880, 14398171200, 12025780892160, 315031400802720, 315031400802720, 12025780892160, 14398171200, 362880, 1

Offset: 0

N. J. A. Sloane

Or, triangle of multipermutation numbers T(n,k), n >= 0, 0 <= k <= n: number of relations on an n-set such that all vertical sections and all horizontal sections have k elements.

			Triangle begins:
  1;
  1,    1;
  1,    2,       1;
  1,    6,       6,        1;
  1,   24,      90,       24,        1;
  1,  120,    2040,     2040,      120,       1;
  1,  720,   67950,   297200,    67950,     720,    1;
  1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,k).

Brendan D. McKay, Rows n = 0..30, flattened
C. J. Everett and P. R. Stein, The asymptotic number of integer stochastic matrices, Disc. Math. 1 (1971), 55-72.
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
Richard J. Mathar, Rencontres for equipartite distributions of multisets of colored balls into urns, vixra:2306.0157 (2023)
B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
Brendan D. McKay, first 30 rows : entries named Bv[n,k,n,k]
Wouter Meeussen, relevant entries from B. D. McKay reference

Row sums give A067209.
Central coefficients are A058527.
Cf. A000142 (column 1), A001499 (column 2), A001501 (column 3), A058528 (column 4), A075754 (column 5), A172544 (column 6), A172541 (column 7), A172536 (column 8), A172540 (column 9), A172535 (column 11), A172534 (column 12), A172538 (column 13), A172537 (column 14).
Cf. A133687, A333157 (symmetric matrices), A257493 (nonnegative elements), A260340 (up to row permutations), A364068 (traceless).

T(n, k)={
  local(M=Map(Mat([n, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(i, p, v, e) = if(i<0, if(!e, acc(p, v)), my(t=polcoef(p,i)); for(j=0, min(t, e), self()(i-1, p+j*(x-1)*x^i, binomial(t, j)*v, e-j))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[,2]);
} \\ Andrew Howroyd, Apr 03 2020

Comtet quotes Everett and Stein as showing that T(n,k) ~ (kn)!(k!)^(-2n) exp( -(k-1)^2/2 ) for fixed k as n -> oo.

T(n,k) = T(n,n-k).

More terms from Greg Kuperberg, Feb 08 2001

A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 20, 90, 20, 1, 1, 70, 1860, 1860, 70, 1, 1, 252, 44730, 297200, 44730, 252, 1, 1, 924, 1172556, 60871300, 60871300, 1172556, 924, 1, 1, 3432, 32496156, 14367744720, 116963796250, 14367744720, 32496156, 3432, 1, 1, 12870, 936369720, 3718394156400, 273957842462220, 273957842462220, 3718394156400, 936369720, 12870, 1

Offset: 0

Andrew Howroyd, Oct 11 2024

T(n,k) is the number of 2*n X 2*k {-1,1} matrices with all rows and columns summing to zero.

			Array begins:
========================================================================
n\k | 0   1       2           3               4                   5 ...
----+------------------------------------------------------------------
  0 | 1   1       1           1               1                   1 ...
  1 | 1   2       6          20              70                 252 ...
  2 | 1   6      90        1860           44730             1172556 ...
  3 | 1  20    1860      297200        60871300         14367744720 ...
  4 | 1  70   44730    60871300    116963796250     273957842462220 ...
  5 | 1 252 1172556 14367744720 273957842462220 6736218287430460752 ...
  ...

Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Main diagonal is A058527.
Columns 0..9 are A000012, A000984, A002896, A172556, A172555, A172557, A172558, A172559, A172560, A172554.
Cf. A008300, A195644, A333901, A334549, A377007 (up to permutations of rows and columns).

T(n, k)={
  local(M=Map(Mat([2*k, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(i, p, v, e) = if(i<0, if(!e, acc(p, v)), my(t=polcoef(p,i)); for(j=0, min(t, e), self()(i-1, p+j*(x-1)*x^i, binomial(t, j)*v, e-j))));
  for(r=1, 2*n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[,2]);
}

T(n,k) = T(k,n).

A253316 Number of 2n X 2n Takuzu grids.

Original entry on oeis.org

1, 2, 72, 4140, 4111116, 48183195384

Offset: 0

Brian Kell, Dec 30 2014

A Takuzu grid is a 2n X 2n zero-one matrix with the following properties:
Every row and every column has n zeros and n ones.
No row or column has three consecutive zeros or three consecutive ones.
All rows are distinct, and all columns are distinct (but a row may be the same as a column).

			The following is a 4 X 4 Takuzu grid:
[ 0  1  1  0 ]
[ 1  0  0  1 ]
[ 0  0  1  1 ]
[ 1  1  0  0 ]

Wikipedia, Takuzu

Cf. A058527.

Number of possible rows equals A003440.

a(5) from Frans J. Faase, Dec 14 2015

cp's OEIS Frontend

A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

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A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.

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A253316 Number of 2n X 2n Takuzu grids.

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Author

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Extensions

cp's OEIS Frontend

A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

Views

Author

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Comments

Examples

References

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Crossrefs

Programs

PARI

Formula

Extensions

A376935 Array read by antidiagonals: T(n,k) is the number of 2*n X 2*k binary matrices with all row sums k and column sums n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A253316 Number of 2n X 2n Takuzu grids.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A376935 Array read by antidiagonals: T(n,k) is the number of 2n X 2k binary matrices with all row sums k and column sums n.