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A331571
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Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.
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12
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1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 8, 23, 0, 0, 1, 1, 16, 290, 184, 0, 0, 1, 1, 32, 4298, 17488, 840, 0, 0, 1, 1, 64, 79143, 2780752, 771305, 0, 0, 0, 1, 1, 128, 1702923, 689187720, 1496866413, 21770070, 0, 0, 0, 1, 1, 256, 42299820, 236477490418, 5261551562405, 585897733896, 328149360, 0, 0, 0, 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))*A331567(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331569(n, j).
A(n, k) = 0 for k > 0, n > 2^(k-1).
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EXAMPLE
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Array begins:
===============================================================
n\k | 0 1 2 3 4 5 6
----+----------------------------------------------------------
0 | 1 1 1 1 1 1 1 ...
1 | 1 1 2 4 8 16 32 ...
2 | 1 0 3 23 290 4298 79143 ...
3 | 1 0 0 184 17488 2780752 689187720 ...
4 | 1 0 0 840 771305 1496866413 5261551562405 ...
5 | 1 0 0 0 21770070 585897733896 30607728081550686 ...
6 | 1 0 0 0 328149360 161088785679360 ...
...
The A(2,2) = 3 matrices are:
[1 1] [1 0] [1 0]
[1 0] [1 1] [0 1]
[0 1] [0 1] [1 1]
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PROG
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(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(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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