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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.
2
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, 79, 67, 37, 14, 5, 0, 6, 68, 170, 184, 116, 55, 19, 6, 0, 8, 122, 407, 543, 417, 219, 91, 28, 8, 0, 10, 232, 1082, 1911, 1760, 1052, 459, 159, 42, 10
OFFSET
0,8
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
EXAMPLE
Triangle begins:
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, 79, 67, 37, 14, 5;
0, 6, 68, 170, 184, 116, 55, 19, 6;
0, 8, 122, 407, 543, 417, 219, 91, 28, 8;
...
The T(4,2) = 5 multiset partitions are:
{{1,1,2,2}}, {{1,2,2,2}}, {{1},{1,2,2}}, {{1},{2,2,2}}, {{1},{1,1,2}}.
These correspond with the following matrices:
[2] [1] [1 1] [1 0] [1 2]
[2] [3] [0 2] [0 3] [0 1]
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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]!))}
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])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}
CROSSREFS
Column k=1 is A000009.
Main diagonal is A000009.
Row sums are A326026.
Cf. A332253.
Sequence in context: A247920 A269735 A187038 * A056619 A324300 A323695
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 08 2020
STATUS
approved