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A331160
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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 distinct nonzero rows with column sums n and rows in decreasing lexicographic order.
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9
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1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 2, 1, 0, 1, 27, 15, 2, 1, 0, 1, 266, 317, 44, 3, 1, 0, 1, 3599, 12586, 2763, 120, 4, 1, 0, 1, 62941, 803764, 390399, 21006, 319, 5, 1, 0, 1, 1372117, 75603729, 103678954, 10074052, 147296, 804, 6, 1
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OFFSET
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0,13
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COMMENTS
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The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows 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)*A219585(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 1 4 27 266 3599 62941 ...
3 | 1 2 15 317 12586 803764 75603729 ...
4 | 1 2 44 2763 390399 103678954 46278915417 ...
5 | 1 3 120 21006 10074052 10679934500 21806685647346 ...
6 | 1 4 319 147296 232165926 956594630508 8717423133548684 ...
7 | 1 5 804 967829 4903530137 76812482919237 ...
...
The A(2,2) = 4 matrices are:
[2 1] [2 0] [1 2] [1 1]
[0 1] [0 2] [1 0] [1 0]
[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+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}
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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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