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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.
11
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
0,13
COMMENTS
The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
LINKS
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).
EXAMPLE
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]
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)/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]))); }
CROSSREFS
Rows 1..3 are A000012, A331704, A331705.
Columns k=0..3 are A000012, A032020, A331706, A331707.
Sequence in context: A119923 A359194 A204420 * A102410 A105123 A058291
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jan 21 2020
STATUS
approved