A331569 Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of distinct nonzero rows with n ones in every column and columns in decreasing lexicographic order.

1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 0, 1, 0, 1, 17, 0, 0, 1, 0, 1, 230, 184, 0, 0, 1, 0, 1, 3264, 16936, 840, 0, 0, 1, 0, 1, 60338, 2711904, 768785, 0, 0, 0, 1, 0, 1, 1287062, 675457000, 1493786233, 21770070, 0, 0, 0, 1, 0, 1, 31900620, 232383728378, 5254074934990, 585810653616, 328149360, 0, 0, 0, 1

Offset

0,13

Comments

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

A(n,k) is the number of k-block n-uniform T_0 set systems without isolated vertices.

Links

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

Formula

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

A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331571(n, j).

A331651(n) = Sum_{d|n} A(n/d, d).

Example

Array begins:
===============================================================
n\k | 0 1 2   3         4               5                 6
----+----------------------------------------------------------
  0 | 1 1 0   0         0               0                 0 ...
  1 | 1 1 1   1         1               1                 1 ...
  2 | 1 0 3  17       230            3264             60338 ...
  3 | 1 0 0 184     16936         2711904         675457000 ...
  4 | 1 0 0 840    768785      1493786233     5254074934990 ...
  5 | 1 0 0   0  21770070    585810653616 30604798810581906 ...
  6 | 1 0 0   0 328149360 161087473081920 ...
  ...
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]

Prog

(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)/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, k<=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]))); }

Crossrefs

Rows n=1..4 are A000012, A331649, A094631, A331650.

Cf. A330942, A331039, A331567, A331570, A331571, A331572, A331651.

Sequence in context: A248678 A325268 A232630 * A341716 A216600 A356679

Adjacent sequences: A331566 A331567 A331568 * A331570 A331571 A331572

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 20 2020

Status

approved