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A331126
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Array read by antidiagonals: A(n,k) is the number of T_0 n-regular set multipartitions (multisets of sets) on a k-set.
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10
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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
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OFFSET
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0,13
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COMMENTS
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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.
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LINKS
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FORMULA
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A(n, k) = Sum_{j=1..k} Stirling1(k, j)*A188392(n, j) for n, k >= 1.
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EXAMPLE
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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}}.
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PROG
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(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]))}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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