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.

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

Author

Andrew Howroyd, Feb 08 2020

Keywords

• nonn
• tabl

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.

Examples

			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]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Column k=1 is A000009.

Main diagonal is A000009.

Row sums are A326026.

Cf. A332253.

Programs

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]))}