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%I #13 Jan 16 2024 17:33:08
%S 1,1,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,5,3,1,1,0,1,11,12,4,1,1,0,1,26,66,
%T 25,5,1,1,0,1,68,445,278,44,6,1,1,0,1,177,4279,5532,966,73,7,1,1,0,1,
%U 497,53340,200589,53535,2957,112,8,1,1,0,1,1476,846254,11662671,7043925,431805,8149,166,9,1,1
%N Array read by antidiagonals: A(n,k) is the number of nonisomorphic 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 nonequivalent binary matrices with k distinct columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.
%C A(n,k) is the number of non-isomorphic set-systems with k parts each of size n.
%H Andrew Howroyd, <a href="/A331508/b331508.txt">Table of n, a(n) for n = 0..152</a> (first 17 antidiagonals)
%F A306019(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 5 11 26 68 177 ...
%e 3 | 1 1 3 12 66 445 4279 53340 ...
%e 4 | 1 1 4 25 278 5532 200589 11662671 ...
%e 5 | 1 1 5 44 966 53535 7043925 ...
%e 6 | 1 1 6 73 2957 431805 ...
%e ...
%e The A(2,3) = 5 matrices are:
%e [1 0 0] [1 1 0] [1 1 1] [1 1 0] [1 1 0]
%e [1 0 0] [1 0 0] [1 0 0] [1 0 1] [1 0 1]
%e [0 1 0] [0 1 0] [0 1 0] [0 1 0] [0 1 1]
%e [0 1 0] [0 0 1] [0 0 1] [0 0 1]
%e [0 0 1] [0 0 1]
%e [0 0 1]
%o (PARI)
%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))[k]}
%o T(n,k)={my(m=n*k, s=0); if(m==0, k<=1, forpart(q=m, my(g=sum(t=1, k, K(q, t, n)*x^t/t) + O(x*x^k)); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), k)); s/m!)}
%o { for(n=0, 6, for(k=0, 5, print1(T(n, k), ", ")); print) } \\ _Andrew Howroyd_, Jan 16 2024
%Y Rows n=1..6 are A000012, A000664, A058790, A058791, A058792, A058793.
%Y Cf. A283877, A306019, A330942, A331126, A331277, A331461, A331509, A331510.
%K nonn,tabl
%O 0,13
%A _Andrew Howroyd_, Jan 18 2020
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