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A331161
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Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and rows in nonincreasing lexicographic order.
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9
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1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 7, 3, 1, 0, 1, 43, 28, 5, 1, 0, 1, 403, 599, 104, 7, 1, 0, 1, 5245, 23243, 6404, 332, 11, 1, 0, 1, 89132, 1440532, 872681, 57613, 1032, 15, 1, 0, 1, 1898630, 131530132, 222686668, 26560747, 473674, 2983, 22, 1
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OFFSET
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0,9
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COMMENTS
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The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows
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LINKS
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FORMULA
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A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219727(n, j).
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EXAMPLE
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Array begins:
====================================================================
n\k | 0 1 2 3 4 5 6
----+---------------------------------------------------------------
0 | 1 1 0 0 0 0 0 ...
1 | 1 1 1 1 1 1 1 ...
2 | 1 2 7 43 403 5245 89132 ...
3 | 1 3 28 599 23243 1440532 131530132 ...
4 | 1 5 104 6404 872681 222686668 95605470805 ...
5 | 1 7 332 57613 26560747 26852940027 52296207431182 ...
6 | 1 11 1032 473674 712725249 2776638423133 ...
7 | 1 15 2983 3599384 17328777789 ...
...
The A(2,2) = 7 matrices are:
[2 1] [2 0] [1 2] [1 1] [2 0] [1 0] [1 0]
[0 1] [0 2] [1 0] [1 0] [0 1] [1 0] [1 0]
[0 1] [0 1] [0 2] [0 1]
[0 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)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, k<=1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p, n, k), [1, n]); s*q[#q-j]))}
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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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