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A330942

Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 7, 1, 1, 1, 8, 75, 32, 1, 1, 1, 16, 1105, 2712, 161, 1, 1, 1, 32, 20821, 449102, 116681, 842, 1, 1, 1, 64, 478439, 122886128, 231522891, 5366384, 4495, 1, 1, 1, 128, 12977815, 50225389432, 975712562347, 131163390878, 256461703, 24320, 1, 1

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OFFSET

0,8

COMMENTS

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

A(n,k) is the number of labeled n-uniform hypergraphs with multiple edges allowed and with k edges and no isolated vertices. When n=2 these objects are multigraphs.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1325

FORMULA

A(n, k) = Sum_{j=0..n*k} binomial(binomial(j,n)+k-1, k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).

A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A262809(n, j)/k!.

A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331277(n, j).

A331638(n) = Sum_{d|n} A(n/d, d).

For fixed r > 1, A(r,n) ~ c * r^(r*n + 1/2) * n^((r-1)*n) / (2^((r+1)/2) * (r!)^n * exp((r-1)*n) * log(2)^(r*n+1)), where c = 2^(log(2)/4) if r=2 and c = 1 if r>2. - Vaclav Kotesovec, Feb 08 2026

EXAMPLE

Array begins:

============================================================

n\k | 0 1 2 3 4 5

----+-------------------------------------------------------

0 | 1 1 1 1 1 1 ...

1 | 1 1 2 4 8 16 ...

2 | 1 1 7 75 1105 20821 ...

3 | 1 1 32 2712 449102 122886128 ...

4 | 1 1 161 116681 231522891 975712562347 ...

5 | 1 1 842 5366384 131163390878 8756434117294432 ...

6 | 1 1 4495 256461703 78650129124911 ...