A067209 -id:A067209 - 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

A202784 T(n,k) is the number of n X n 0..k arrays with row and column sums equal.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 14, 1, 1, 5, 16, 87, 140, 1, 1, 6, 25, 340, 5673, 4322, 1, 1, 7, 36, 1001, 89520, 2577513, 434542, 1, 1, 8, 49, 2442, 790425, 290838284, 8459809773, 144109562, 1, 1, 9, 64, 5215, 4756140, 11991502235, 12092687195284, 207495141083975, 165431317452, 1

Offset: 0

R. H. Hardin, Dec 24 2011

			Table starts
n\k | 0            1               2              3                4
----+-------------------------------------------------------------------
  0 | 1            1               1              1                1 ...
  1 | 1            2               3              4                5 ...
  2 | 1            4               9             16               25 ...
  3 | 1           14              87            340             1001 ...
  4 | 1          140            5673          89520           790425 ...
  5 | 1         4322         2577513      290838284      11991502235 ...
  6 | 1       434542      8459809773 12092687195284 3632894572698505 ...
  7 | 1    144109562 207495141083975 ...
  8 | 1 165431317452 ...
  ...
Some solutions for n=4 k=3
..3..1..2..1....2..2..0..3....1..1..1..2....1..3..0..0....1..2..0..2
..2..3..2..0....1..2..2..2....3..0..1..1....1..0..1..2....3..1..0..1
..2..2..0..3....3..0..3..1....0..3..2..0....2..0..1..1....0..0..3..2
..0..1..3..3....1..3..2..1....1..1..1..2....0..1..2..1....1..2..2..0

Andrew Howroyd, Table of n, a(n) for n = 0..119 (first 15 antidiagonals; terms up to n=61 from R. H. Hardin)

Rows n=0..5 are A000012, A000027(k+1), A000290(k+1), A202785, A202786, A202787.
Columns k=1..15 are A067209, A067210, A067211, A067212, A067213, A067214, A067215, A067216, A067217, A067218, A067219, A067220, A067221, A067222, A067223.
Cf. A008300, A257493.

Added crossrefs. - R. H. Hardin, Jan 05 2012

Row and column zero inserted by Andrew Howroyd, Oct 14 2024

A181236 T(n,k)=Number of (k*n)Xn binary matrices with all row sums equal and all column sums equal.

Original entry on oeis.org

2, 2, 4, 2, 8, 14, 2, 22, 182, 140, 2, 72, 3362, 49772, 4322, 2, 254, 69302, 33235358, 113400002, 434542, 2, 926, 1513514, 27896484332, 5210265060002, 4027811102702, 144109562, 2, 3434, 34306274, 26012734507190, 298289608088472002

Offset: 1

R. H. Hardin Oct 10 2010

Table starts
............2...................2....................2..................2
............4...................8...................22.................72
...........14.................182.................3362..............69302
..........140...............49772.............33235358........27896484332
.........4322...........113400002........5210265060002.298289608088472002
.......434542.......4027811102702.90698868503010138802...................
....144109562.1237505791330809002........................................
.165431317452............................................................

			All solutions for 6X2
..0..0....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..0..0....0..1....0..1....0..1....0..1....1..0....1..0....1..0....1..0....1..0
..0..0....0..1....1..0....1..0....1..0....0..1....0..1....0..1....1..0....1..0
..0..0....1..0....0..1....1..0....1..0....0..1....1..0....1..0....0..1....0..1
..0..0....1..0....1..0....1..0....0..1....1..0....1..0....0..1....1..0....0..1
..0..0....1..0....1..0....0..1....1..0....1..0....0..1....1..0....0..1....1..0
...
..0..1....1..0....1..0....1..0....1..0....1..0....1..0....1..0....1..0....1..0
..1..0....0..1....0..1....0..1....0..1....0..1....0..1....1..0....1..0....1..0
..1..0....0..1....0..1....0..1....1..0....1..0....1..0....0..1....0..1....0..1
..1..0....0..1....1..0....1..0....0..1....0..1....1..0....0..1....0..1....1..0
..0..1....1..0....1..0....0..1....1..0....0..1....0..1....1..0....0..1....0..1
..0..1....1..0....0..1....1..0....0..1....1..0....0..1....0..1....1..0....0..1
...
..1..0....1..1
..1..0....1..1
..1..0....1..1
..0..1....1..1
..0..1....1..1
..0..1....1..1

R. H. Hardin, Table of n, a(n) for n=1..41

Column 1 is A067209

Row 2 is twice A112849

A333681 Number of non-isomorphic n X n binary matrices with all row and column sums equal up to permutation of rows and columns.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 19, 44, 314, 7526, 846993, 324127860, 403254094632, 1555631972009430, 19731915624463099553, 791773335030637885025288, 107432353216118868234728540268, 47049030539260648478475949282317452, 71364337698829887974206671525372672234855

Offset: 0

Andrew Howroyd, Apr 01 2020

nonn

			The a(2) = 3 matrices are:
  [0 0]  [0 1]  [1 1]
  [0 0]  [1 0]  [1 1]

Row sums of A133687.

Cf. A000519, A067209, A333160.

a(n) = A000519(n) + 1.

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

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A202784 T(n,k) is the number of n X n 0..k arrays with row and column sums equal.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A181236 T(n,k)=Number of (k*n)Xn binary matrices with all row sums equal and all column sums equal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A333681 Number of non-isomorphic n X n binary matrices with all row and column sums equal up to permutation of rows and columns.

Views

Author

Keywords

Examples

Crossrefs

Formula