A331570 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of distinct nonzero rows with column sums n and columns in decreasing lexicographic order.

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 6, 3, 1, 0, 1, 46, 42, 3, 1, 0, 1, 544, 1900, 268, 5, 1, 0, 1, 7983, 184550, 73028, 1239, 11, 1, 0, 1, 144970, 29724388, 57835569, 2448599, 7278, 13, 1, 0, 1, 3097825, 7137090958, 99940181999, 16550232235, 75497242, 40828, 19, 1

Offset: 0

Author

Andrew Howroyd, Jan 21 2020

Keywords

• nonn
• tabl

Comments

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

Examples

			Array begins:
=============================================================
n\k | 0  1    2        3             4                  5
----+--------------------------------------------------------
  0 | 1  1    0        0             0                  0 ...
  1 | 1  1    1        1             1                  1 ...
  2 | 1  1    6       46           544               7983 ...
  3 | 1  3   42     1900        184550           29724388 ...
  4 | 1  3  268    73028      57835569        99940181999 ...
  5 | 1  5 1239  2448599   16550232235    311353753947045 ...
  6 | 1 11 7278 75497242 4388476386528 896320470282357104 ...
  ...
The A(2,2) = 6 matrices are:
   [1 1]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]
   [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 1..3 are A000012, A331704, A331705.

Columns k=0..3 are A000012, A032020, A331706, A331707.

Cf. A331315, A331568, A331569, A331571, A331572, A331708.

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)/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, k<=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} Stirling1(k, j)*A331568(n, j)/k!.

A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331572(n, j).

A331708(n) = Sum_{d|n} A(n/d, d).