A369287 Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n with k parts and no singletons or vertices that appear only once, 0 <= k <= floor(n/2).

1, 0, 0, 1, 0, 1, 0, 2, 3, 0, 2, 4, 0, 4, 15, 8, 0, 4, 24, 19, 0, 7, 60, 79, 23, 0, 8, 101, 213, 84, 0, 12, 210, 615, 424, 66, 0, 14, 357, 1523, 1533, 363, 0, 21, 679, 3851, 5580, 2217, 212, 0, 24, 1142, 8963, 17836, 10379, 1575, 0, 34, 2049, 20840, 55730, 45866, 11616, 686

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 sum at least two up to permutation of rows and columns.

Examples

			Triangle begins:
  1;
  0;
  0,  1;
  0,  1;
  0,  2,    3;
  0,  2,    4;
  0,  4,   15,    8;
  0,  4,   24,   19;
  0,  7,   60,   79,    23;
  0,  8,  101,  213,    84;
  0, 12,  210,  615,   424,    66;
  0, 14,  357, 1523,  1533,   363;
  0, 21,  679, 3851,  5580,  2217,  212;
  0, 24, 1142, 8963, 17836, 10379, 1575;
  ...
The T(5,1) = 2 multiset partitions are:
   {{1,1,1,1,1}},
   {{1,1,1,2,2}}.
The corresponding T(5,1) = 2 matrices are:
   [5]  [3 2].
The T(5,2) = 4 matrices are:
   [3]  [3 0]  [2 1]  [2 1]
   [2]  [0 2]  [1 1]  [0 2],
		

Links

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

Crossrefs

Row sums are A320665.

Columns k=0..1 are A000007, A002865(n>0).

Cf. A050535, A369286.

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]))); (1 - x)*x*Ser(K(q,t,k)) - c*x}
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) = A050535(n).