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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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%I #11 Jan 24 2020 15:53:39

%S 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,

%T 3599,12586,2763,120,4,1,0,1,62941,803764,390399,21006,319,5,1,0,1,

%U 1372117,75603729,103678954,10074052,147296,804,6,1

%N 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.

%C The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.

%H Andrew Howroyd, <a href="/A331160/b331160.txt">Table of n, a(n) for n = 0..209</a>

%F A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219585(n, j).

%F A331318(n) = Sum_{d|n} A(n/d, d).

%e Array begins:

%e ===================================================================

%e n\k | 0 1 2 3 4 5 6

%e ----+--------------------------------------------------------------

%e 0 | 1 1 0 0 0 0 0 ...

%e 1 | 1 1 1 1 1 1 1 ...

%e 2 | 1 1 4 27 266 3599 62941 ...

%e 3 | 1 2 15 317 12586 803764 75603729 ...

%e 4 | 1 2 44 2763 390399 103678954 46278915417 ...

%e 5 | 1 3 120 21006 10074052 10679934500 21806685647346 ...

%e 6 | 1 4 319 147296 232165926 956594630508 8717423133548684 ...

%e 7 | 1 5 804 967829 4903530137 76812482919237 ...

%e ...

%e The A(2,2) = 4 matrices are:

%e [2 1] [2 0] [1 2] [1 1]

%e [0 1] [0 2] [1 0] [1 0]

%e [0 1]

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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]!)}

%o 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)}

%Y Rows n=1..3 are A000012, A331316, A331344

%Y Columns k=0..2 are A000012, A000009, A331317.

%Y Cf. A219585, A331039, A331126, A331161, A331315, A331318.

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, Jan 10 2020