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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. 10
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

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.

LINKS

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

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).

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 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}}.

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, 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]))}

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.

Sequence in context: A275784 A331508 A097608 * A168261 A180997 A143439

Adjacent sequences:  A331123 A331124 A331125 * A331127 A331128 A331129

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Jan 10 2020

STATUS

approved

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Last modified July 14 09:22 EDT 2020. Contains 335720 sequences. (Running on oeis4.)