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

Author

N. J. A. Sloane

Keywords

• tabl
• nonn
• nice

Comments

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.

Examples

			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;
  ...
		

References

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

Links

• 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

Crossrefs

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).

Programs

PARI

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

Formula

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).

Extensions

More terms from Greg Kuperberg, Feb 08 2001