A331039 Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set-systems on a k-set.

1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 5, 0, 0, 1, 0, 1, 43, 5, 0, 0, 1, 0, 1, 518, 175, 1, 0, 0, 1, 0, 1, 8186, 9426, 272, 0, 0, 0, 1, 0, 1, 163356, 751365, 64453, 205, 0, 0, 0, 1, 0, 1, 3988342, 84012191, 23553340, 248685, 80, 0, 0, 0, 1

Offset: 0

Author

Andrew Howroyd, Jan 08 2020

Keywords

• nonn
• tabl

Comments

An n-regular set-system is a finite set of nonempty sets in which each element appears in n blocks.

A set-system is T_0 if for every two distinct elements there exists a block containing one but not the other element.

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 rows in decreasing lexicographic order.

Examples

			Array begins:
==========================================================
n\k | 0 1 2 3   4       5           6                7
----+-----------------------------------------------------
  0 | 1 1 0 0   0       0           0                0 ...
  1 | 1 1 1 1   1       1           1                1 ...
  2 | 1 0 1 5  43     518        8186           163356 ...
  3 | 1 0 0 5 175    9426      751365         84012191 ...
  4 | 1 0 0 1 272   64453    23553340      13241130441 ...
  5 | 1 0 0 0 205  248685   421934358    1176014951129 ...
  6 | 1 0 0 0  80  620548  5055634889   69754280936418 ...
  7 | 1 0 0 0  15 1057989 43402628681 2972156676325398 ...
  ...
The A(2,3) = 5 matrices are:
  [1 1 1]    [1 1 0]    [1 1 0]    [1 0 1]    [1 1 0]
  [1 0 0]    [1 0 1]    [1 0 0]    [1 0 0]    [1 0 1]
  [0 1 0]    [0 1 0]    [0 1 1]    [0 1 1]    [0 1 1]
  [0 0 1]    [0 0 1]    [0 0 1]    [0 1 0]
The corresponding set-systems are:
  {{1,2,3}, {1}, {2}, {3}},
  {{1,2}, {1,3}, {2,3}},
  {{1,2}, {1,3}, {2}, {3}},
  {{1,2}, {1}, {2,3}, {3}},
  {{1,3}, {1}, {2,3}, {2}}.
		

Links

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

Crossrefs

Rows n=1..4 are A000012, A060053, A060070, A331655.

Cf. A188445, A330942, A330964, A331126, A331160, A331161, A331569, A331654.

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)*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)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}

Formula

A(n, k) = Sum_{j=1..k} Stirling1(k, j)*A188445(n, j) for n, k >= 1.

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

A331654(n) = Sum_{d|n} A(n/d, d).