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

0,8

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

Adjacent sequences: A332257 A332258 A332259 * A332261 A332262 A332263

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 08 2020

Status

approved