A331571 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 8, 23, 0, 0, 1, 1, 16, 290, 184, 0, 0, 1, 1, 32, 4298, 17488, 840, 0, 0, 1, 1, 64, 79143, 2780752, 771305, 0, 0, 0, 1, 1, 128, 1702923, 689187720, 1496866413, 21770070, 0, 0, 0, 1, 1, 256, 42299820, 236477490418, 5261551562405, 585897733896, 328149360, 0, 0, 0, 1

Offset: 0

Author

Andrew Howroyd, Jan 20 2020

Keywords

• nonn
• tabl

Comments

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

Examples

			Array begins:
===============================================================
n\k | 0 1 2   3         4               5                 6
----+----------------------------------------------------------
  0 | 1 1 1   1         1               1                 1 ...
  1 | 1 1 2   4         8              16                32 ...
  2 | 1 0 3  23       290            4298             79143 ...
  3 | 1 0 0 184     17488         2780752         689187720 ...
  4 | 1 0 0 840    771305      1496866413     5261551562405 ...
  5 | 1 0 0   0  21770070    585897733896 30607728081550686 ...
  6 | 1 0 0   0 328149360 161088785679360 ...
  ...
The A(2,2) = 3 matrices are:
   [1 1]  [1 0]  [1 0]
   [1 0]  [1 1]  [0 1]
   [0 1]  [0 1]  [1 1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..209

Crossrefs

Rows n=0..4 are A000012, A011782, A060090, A060491, A331652.

Cf. A330942, A331567, A331569, A331570, A331572, A331653.

Programs

PARI

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n] + k - 1, k)/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

Formula

A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A331567(n, j)/k!.

A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331569(n, j).

A(n, k) = 0 for k > 0, n > 2^(k-1).

A331653(n) = Sum_{d|n} A(n/d, d).