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

1,3

Comments

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

Links

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

Example

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}}.

Prog

(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])) }

Crossrefs

Row sums are A049311.

Main diagonal is A000041.

Cf. A317533, A321609.

Sequence in context: A078032 A162453 A008313 * A232177 A111377 A014046

Adjacent sequences: A334547 A334548 A334549 * A334551 A334552 A334553

Keyword

nonn,tabl

Author

Andrew Howroyd, Jul 03 2020

Status

approved