A257493 Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 21, 24, 1, 1, 1, 5, 55, 282, 120, 1, 1, 1, 6, 120, 2008, 6210, 720, 1, 1, 1, 7, 231, 10147, 153040, 202410, 5040, 1, 1, 1, 8, 406, 40176, 2224955, 20933840, 9135630, 40320, 1, 1, 1, 9, 666, 132724, 22069251, 1047649905, 4662857360, 545007960, 362880, 1

Offset

0,9

Comments

Also the number of ordered factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity). A(2,2) = 3: (2*3)^2 = 36 = 4*9 = 6*6 = 9*4.

Links

Alois P. Heinz, Antidiagonals n = 0..20, flattened

E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)

Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.

Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9

M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]

Example

Square array A(n,k) begins:
  1,   1,      1,        1,          1,           1,            1, ...
  1,   1,      1,        1,          1,           1,            1, ...
  1,   2,      3,        4,          5,           6,            7, ...
  1,   6,     21,       55,        120,         231,          406, ...
  1,  24,    282,     2008,      10147,       40176,       132724, ...
  1, 120,   6210,   153040,    2224955,    22069251,    164176640, ...
  1, 720, 202410, 20933840, 1047649905, 30767936616, 602351808741, ...

Maple

with(numtheory):
b:= proc(n, k) option remember; `if`(n=1, 1, add(
      `if`(bigomega(d)=k, b(n/d, k), 0), d=divisors(n)))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

Mathematica

b[n_, k_] := b[n, k] = If[n==1, 1, Sum[If[PrimeOmega[d]==k, b[n/d, k], 0], {d, Divisors[n]}]]; A[n_, k_] := b[Product[Prime[i], {i, 1, n}]^k, k]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

Prog

(Sage)
bigomega = sloane.A001222
@cached_function
def b(n, k):
    if n == 1:
        return 1
    return sum(b(n//d, k) if bigomega(d) == k else 0 for d in n.divisors())
def A(n, k):
    return b(prod(nth_prime(i) for i in (1..n))^k, k)
[A(n, d-n) for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018, translated from Maple
(Sage)
from sage.combinat.integer_matrices import IntegerMatrices
[IntegerMatrices([d-n]*n, [d-n]*n).cardinality() for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018
(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(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(k, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, e\m), self()(if(j==t, k, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k, src[i, 1], 0, src[i, 2], k))); vecsum(Mat(M)[, 2])
} \\ Andrew Howroyd, Apr 04 2020

Crossrefs

Columns k=0-9 give: A000012, A000142, A000681, A001500, A172806, A172862, A172894, A172919, A172944, A172958.

Rows n=0+1, 2-9 give: A000012, A000027(k+1), A002817(k+1), A001496, A003438, A003439, A008552, A160318, A160319.

Main diagonal gives A110058.

Cf. A257463 (unordered factorizations), A333733 (non-isomorphic matrices), A008300 (binary matrices).

Sequence in context: A291709 A326323 A368119 * A296526 A362377 A259844

Adjacent sequences: A257490 A257491 A257492 * A257494 A257495 A257496

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 26 2015

Status

approved