A369286 Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n with k parts and no constant parts or vertices that appear in only one part, 0 <= k <= floor(n/2).

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 5, 2, 0, 0, 6, 3, 0, 0, 16, 16, 5, 0, 0, 22, 44, 13, 0, 0, 45, 135, 82, 11, 0, 0, 64, 338, 301, 52, 0, 0, 119, 880, 1233, 382, 34, 0, 0, 171, 2024, 4090, 1936, 211, 0, 0, 294, 4674, 13474, 9500, 1843, 87, 0, 0, 433, 10191, 40532, 40817, 11778, 873

Offset: 0

Author

Andrew Howroyd, Jan 28 2024

Keywords

• nonn
• tabf

Comments

T(n,k) is the number of nonnegative integer matrices with sum of values n, k rows and every row and column having at least two nonzero entries up to permutation of rows and columns.

Examples

			Triangle begins:
  1;
  0;
  0, 0;
  0, 0;
  0, 0,   1;
  0, 0,   1;
  0, 0,   5,    2;
  0, 0,   6,    3;
  0, 0,  16,   16,    5;
  0, 0,  22,   44,   13;
  0, 0,  45,  135,   82,   11;
  0, 0,  64,  338,  301,   52;
  0, 0, 119,  880, 1233,  382,  34;
  0, 0, 171, 2024, 4090, 1936, 211;
  ...
The T(6,2) = 5 multiset partitions are:
  {{1,1,1,2}, {1,2}},
  {{1,1,2,2}, {1,2}},
  {{1,1,2}, {1,1,2}},
  {{1,1,2}, {1,2,2}},
  {{1,2,3}, {1,2,3}}.
The corresponding T(6,2) = 5 matrices are:
  [3 1]  [2 2]  [2 1]  [2 1]  [1 1 1]
  [1 1]  [1 1]  [2 1]  [1 2]  [1 1 1]
The T(6,3) = 2 matrices are:
  [1 1]  [1 1 0]
  [1 1]  [1 0 1]
  [1 1]  [0 1 1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..675 (rows 0..50)

Crossrefs

Row sums are A321760.

Cf. A307316, A369287.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); eta(x + O(x*x^k))*(1 + x*Ser(K(q,t,k))) + x*(1-c)/(1-x) - 1}
G(n,y=1)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, subst(H(q, t, n\t)*y^t/t, x, x^t) ))); s/n!}
T(n)={my(v=Vec(G(n,'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
{ my(A=T(15)); for(i=1, #A, print(A[i])) }

Formula

T(2*n,n) = A307316(n).