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A331039 Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set-systems on a k-set. 14
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,18

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.

LINKS

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

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).

EXAMPLE

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}}.

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)*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)}

CROSSREFS

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

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

Sequence in context: A047754 A048682 A186716 * A171915 A287703 A316480

Adjacent sequences:  A331036 A331037 A331038 * A331040 A331041 A331042

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Jan 08 2020

STATUS

approved

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Last modified July 27 07:36 EDT 2021. Contains 346304 sequences. (Running on oeis4.)