A334550 Triangle read by rows: T(n,k) is the number of binary matrices with n ones, k columns and no zero rows or columns, up to permutations of rows and columns.

1, 1, 2, 1, 2, 3, 1, 5, 5, 5, 1, 5, 12, 9, 7, 1, 9, 23, 29, 17, 11, 1, 9, 39, 62, 57, 28, 15, 1, 14, 63, 147, 154, 110, 47, 22, 1, 14, 102, 278, 409, 329, 194, 73, 30, 1, 20, 150, 568, 991, 1023, 664, 335, 114, 42, 1, 20, 221, 1020, 2334, 2844, 2267, 1243, 549, 170, 56

Offset: 1

Author

Andrew Howroyd, Jul 03 2020

Keywords

• nonn
• tabl

Comments

T(n,k) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts.

Examples

			Triangle begins:
  1;
  1,  2;
  1,  2,  3;
  1,  5,  5,   5;
  1,  5, 12,   9,   7;
  1,  9, 23,  29,  17,  11;
  1,  9, 39,  62,  57,  28, 15;
  1, 14, 63, 147, 154, 110, 47, 22;
  ...
The T(4,3) = 5 matrices are:
  [1 0 0]   [1 0 0]   [1 1 0]   [1 1 1]   [1 1 0]
  [1 0 0]   [1 0 0]   [1 0 0]   [1 0 0]   [1 0 1]
  [0 1 0]   [0 1 1]   [0 0 1]
  [0 0 1]
The T(4,3) = 5 the set multipartitions are:
  {{1,2}, {3}, {4}},
  {{1,2}, {3}, {3}},
  {{1,2}, {1}, {3}},
  {{1,2}, {1}, {1}},
  {{1,2}, {1}, {2}}.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums are A049311.

Main diagonal is A000041.

Cf. A317533, A321609.

Programs

PARI

\\ See A321609 for definition of M.
T(n, k)={M(k, n, n) - M(k-1, n, n)}
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print)

PARI

\\ Faster version.
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, n)={prod(j=1, #q, (1+x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}
G(m,n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}
A(n,m=n)={my(p=sum(k=0, m, G(k,n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}
{ my(T=A(10)); for(n=1, #T, print(T[n,1..n])) }