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%I #15 Jan 25 2020 17:54:55
%S 1,1,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,9,3,1,1,0,1,70,29,4,1,1,0,1,794,
%T 666,68,5,1,1,0,1,12055,28344,3642,134,6,1,1,0,1,233238,1935054,
%U 469368,14951,237,7,1,1,0,1,5556725,193926796,119843417,5289611,50985,388,8,1,1
%N Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set multipartitions (multisets of sets) on a k-set.
%C An n-regular set multipartition is a finite multiset of nonempty sets in which each element appears in n blocks.
%C A set multipartition is T_0 if for every two distinct elements there exists a block containing one but not the other element.
%C A(n,k) is the number of binary matrices with k distinct columns and any number of nonzero rows with n ones in every column and rows in nonincreasing lexicographic order.
%H Andrew Howroyd, <a href="/A331126/b331126.txt">Table of n, a(n) for n = 0..209</a>
%F A(n, k) = Sum_{j=1..k} Stirling1(k, j)*A188392(n, j) for n, k >= 1.
%F A331391(n) = Sum_{d|n} A(n/d, d).
%e Array begins:
%e ====================================================================
%e n\k | 0 1 2 3 4 5 6 7
%e ----+---------------------------------------------------------------
%e 0 | 1 1 0 0 0 0 0 0 ...
%e 1 | 1 1 1 1 1 1 1 1 ...
%e 2 | 1 1 2 9 70 794 12055 233238 ...
%e 3 | 1 1 3 29 666 28344 1935054 193926796 ...
%e 4 | 1 1 4 68 3642 469368 119843417 53059346010 ...
%e 5 | 1 1 5 134 14951 5289611 4681749424 8639480647842 ...
%e 6 | 1 1 6 237 50985 46241343 134332244907 989821806791367 ...
%e 7 | 1 1 7 388 151901 333750928 3032595328876 85801167516707734 ...
%e ...
%e The A(2,2) = 2 matrices are:
%e [1 1] [1 0]
%e [1 0] [1 0]
%e [0 1] [0 1]
%e [0 1]
%e The corresponding set multipartitions are:
%e {{1,2}, {1}, {2}},
%e {{1}, {1}, {2}, {2}}.
%o (PARI)
%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
%o 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]))}
%Y Rows n=1..3 are A000012, A014500, A331389.
%Y Columns k=0..3 are A000012, A000012, A001477, A331390.
%Y Cf. A188392, A188445, A330942, A331039, A331160, A331161, A331391.
%K nonn,tabl
%O 0,13
%A _Andrew Howroyd_, Jan 10 2020
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