A331126 Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set multipartitions (multisets of sets) on a k-set.

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 9, 3, 1, 1, 0, 1, 70, 29, 4, 1, 1, 0, 1, 794, 666, 68, 5, 1, 1, 0, 1, 12055, 28344, 3642, 134, 6, 1, 1, 0, 1, 233238, 1935054, 469368, 14951, 237, 7, 1, 1, 0, 1, 5556725, 193926796, 119843417, 5289611, 50985, 388, 8, 1, 1

Offset: 0

Author

Andrew Howroyd, Jan 10 2020

Keywords

• nonn
• tabl

Comments

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

A set multipartition 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 nonzero rows with n ones in every column and rows in nonincreasing 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 1 2   9     70       794         12055            233238 ...
  3 | 1 1 3  29    666     28344       1935054         193926796 ...
  4 | 1 1 4  68   3642    469368     119843417       53059346010 ...
  5 | 1 1 5 134  14951   5289611    4681749424     8639480647842 ...
  6 | 1 1 6 237  50985  46241343  134332244907   989821806791367 ...
  7 | 1 1 7 388 151901 333750928 3032595328876 85801167516707734 ...
     ...
The A(2,2) = 2 matrices are:
   [1 1]   [1 0]
   [1 0]   [1 0]
   [0 1]   [0 1]
           [0 1]
The corresponding set multipartitions are:
    {{1,2}, {1}, {2}},
    {{1}, {1}, {2}, {2}}.
		

Links

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

Crossrefs

Rows n=1..3 are A000012, A014500, A331389.

Columns k=0..3 are A000012, A000012, A001477, A331390.

Cf. A188392, A188445, A330942, A331039, A331160, A331161, A331391.

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

Formula

A(n, k) = Sum_{j=1..k} Stirling1(k, j)*A188392(n, j) for n, k >= 1.

A331391(n) = Sum_{d|n} A(n/d, d).