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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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified February 26 14:18 EST 2021. Contains 341632 sequences. (Running on oeis4.)