A331572 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of distinct nonzero rows with column sums n and columns in nonincreasing lexicographic order.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 7, 3, 1, 1, 8, 59, 45, 3, 1, 1, 16, 701, 1987, 271, 5, 1, 1, 32, 10460, 190379, 73567, 1244, 11, 1, 1, 64, 190816, 30474159, 58055460, 2451082, 7289, 13, 1, 1, 128, 4098997, 7287577611, 100171963518, 16557581754, 75511809, 40841, 19, 1

Offset: 0

Author

Andrew Howroyd, Jan 21 2020

Keywords

• nonn
• tabl

Comments

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

Examples

			Array begins:
==========================================================
n\k | 0  1    2        3             4               5
----+-----------------------------------------------------
  0 | 1  1    1        1             1               1 ...
  1 | 1  1    2        4             8              16 ...
  2 | 1  1    7       59           701           10460 ...
  3 | 1  3   45     1987        190379        30474159 ...
  4 | 1  3  271    73567      58055460    100171963518 ...
  5 | 1  5 1244  2451082   16557581754 311419969572540 ...
  6 | 1 11 7289 75511809 4388702900099 ...
  ...
The A(2,2) = 7 matrices are:
   [1 1]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]  [2 2]
   [1 0]  [1 1]  [0 1]  [0 1]  [0 2]  [1 2]
   [0 1]  [0 1]  [1 1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..209

Crossrefs

Rows n=0..3 are A000012, A011782, A331709, A331710.

Columns k=0..3 are A000012, A032020, A331711, A331712.

Cf. A331315, A331568, A331569, A331570, A331571, A331713.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n] + k - 1, k)/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

Formula

A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A331568(n, j)/k!.

A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331570(n, j).

A331713(n) = Sum_{d|n} A(n/d, d).