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

0,5

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.

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

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).

Example

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;
  ...

Prog

(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

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).

Sequence in context: A376935 A155795 A009963 * A321789 A173887 A288025

Adjacent sequences: A008297 A008298 A008299 * A008301 A008302 A008303

Keyword

tabl,nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Greg Kuperberg, Feb 08 2001

Status

approved