cp's OEIS Frontend

A332260 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.

%I A332260 #11 Dec 23 2020 01:51:18
%S A332260 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 A332260 79,67,37,14,5,0,6,68,170,184,116,55,19,6,0,8,122,407,543,417,219,91,
%U A332260 28,8,0,10,232,1082,1911,1760,1052,459,159,42,10
%N A332260 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 A332260 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 A332260 Andrew Howroyd, <a href="/A332260/b332260.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A332260 Triangle begins:
%e A332260   1;
%e A332260   0, 1;
%e A332260   0, 1,   1;
%e A332260   0, 2,   3,   2;
%e A332260   0, 2,   5,   3,   2;
%e A332260   0, 3,  11,  12,   6,   3;
%e A332260   0, 4,  26,  39,  27,  11,   4;
%e A332260   0, 5,  40,  79,  67,  37,  14,  5;
%e A332260   0, 6,  68, 170, 184, 116,  55, 19,  6;
%e A332260   0, 8, 122, 407, 543, 417, 219, 91, 28, 8;
%e A332260   ...
%e A332260 The T(4,2) = 5 multiset partitions are:
%e A332260   {{1,1,2,2}}, {{1,2,2,2}}, {{1},{1,2,2}}, {{1},{2,2,2}}, {{1},{1,1,2}}.
%e A332260 These correspond with the following matrices:
%e A332260    [2]  [1]  [1 1]  [1 0]  [1 2]
%e A332260    [2]  [3]  [0 2]  [0 3]  [0 1]
%o A332260 (PARI)
%o A332260 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A332260 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 A332260 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 A332260 {my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}
%Y A332260 Column k=1 is A000009.
%Y A332260 Main diagonal is A000009.
%Y A332260 Row sums are A326026.
%Y A332260 Cf. A332253.
%K A332260 nonn,tabl
%O A332260 0,8
%A A332260 _Andrew Howroyd_, Feb 08 2020