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

0,8

Comments

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

Links

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

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).

Example

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]

Prog

(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]))); }

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.

Sequence in context: A295685 A330942 A141471 * A127080 A216645 A216635

Adjacent sequences: A331569 A331570 A331571 * A331573 A331574 A331575

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 21 2020

Status

approved