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A331572
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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.
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11
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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
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OFFSET
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0,8
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COMMENTS
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The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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]
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PROG
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(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]))); }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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