A369927 Triangle read by rows: T(n,k) is the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts and no singletons or endpoints, 0 <= k <= floor(n/2).

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 3, 5, 0, 0, 0, 3, 5, 0, 0, 1, 5, 17, 11, 0, 0, 0, 4, 20, 21, 0, 0, 1, 9, 53, 80, 34, 0, 0, 0, 6, 60, 167, 91, 0, 0, 1, 11, 121, 418, 410, 87, 0, 0, 0, 10, 149, 816, 1189, 402

Offset: 0

Author

Andrew Howroyd, Feb 06 2024

Keywords

• nonn
• tabf

Comments

A singleton is a part of size 1. An endpoint is a vertex that appears in only one part.

T(n,k) is the number of binary matrices with n 1's, k rows and every row and column sum at least two up to permutation of rows and columns.

Examples

			Triangle begins:
  1;
  0;
  0, 0;
  0, 0;
  0, 0, 1;
  0, 0, 0;
  0, 0, 1,  2;
  0, 0, 0,  1;
  0, 0, 1,  3,   5;
  0, 0, 0,  3,   5;
  0, 0, 1,  5,  17,  11;
  0, 0, 0,  4,  20,  21;
  0, 0, 1,  9,  53,  80,   34;
  0, 0, 0,  6,  60, 167,   91;
  0, 0, 1, 11, 121, 418,  410,  87;
  0, 0, 0, 10, 149, 816, 1189, 402;
  ...
The T(4,2) = 1 partition is {{1,2},{1,2}}.
The corresponding matrix is:
   [1 1]
   [1 1]
The T(8,3) = 3 matrices are:
   [1 1 1]  [1 1 1 0]  [1 1 1 1]
   [1 1 1]  [1 1 0 1]  [1 1 0 0]
   [1 1 0]  [0 0 1 1]  [0 0 1 1]
		

Links

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

Crossrefs

Row sums are A369926.

Cf. A307316, A369286, A369287.

Programs

PARI

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(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)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g}
H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); 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) = A307316(n).