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%I #11 Dec 23 2020 01:51:18
%S 1,0,1,0,1,1,0,2,3,2,0,2,5,3,2,0,3,11,12,6,3,0,4,26,39,27,11,4,0,5,40,
%T 79,67,37,14,5,0,6,68,170,184,116,55,19,6,0,8,122,407,543,417,219,91,
%U 28,8,0,10,232,1082,1911,1760,1052,459,159,42,10
%N Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n whose union is a k-set where each part has a different size.
%C T(n,k) is the number of nonequivalent nonnegative integer matrices with total sum n and k nonzero rows with distinct column sums up to permutation of rows and columns.
%H Andrew Howroyd, <a href="/A332260/b332260.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 3, 2;
%e 0, 2, 5, 3, 2;
%e 0, 3, 11, 12, 6, 3;
%e 0, 4, 26, 39, 27, 11, 4;
%e 0, 5, 40, 79, 67, 37, 14, 5;
%e 0, 6, 68, 170, 184, 116, 55, 19, 6;
%e 0, 8, 122, 407, 543, 417, 219, 91, 28, 8;
%e ...
%e The T(4,2) = 5 multiset partitions are:
%e {{1,1,2,2}}, {{1,2,2,2}}, {{1},{1,2,2}}, {{1},{2,2,2}}, {{1},{1,1,2}}.
%e These correspond with the following matrices:
%e [2] [1] [1 1] [1 0] [1 2]
%e [2] [3] [0 2] [0 3] [0 1]
%o (PARI)
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o D(p,n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); prod(j=1, #u, 1 + u[j]*x^j + O(x*x^n))/if(!#p, 1, prod(i=1, p[#p], i^v[i]*v[i]!))}
%o M(n)={my(v=vector(n+1)); for(i=0, n, my(s=0); forpart(p=i, s+=D(p,n)); v[1+i]=Col(s)); Mat(vector(#v, i, v[i]-if(i>1, v[i-1])))}
%o {my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}
%Y Column k=1 is A000009.
%Y Main diagonal is A000009.
%Y Row sums are A326026.
%Y Cf. A332253.
%K nonn,tabl
%O 0,8
%A _Andrew Howroyd_, Feb 08 2020
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