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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. 21
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 (list; table; graph; refs; listen; history; text; internal format)
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).

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 ...

  ...

The A(2,2) = 7 matrices are:

   [1 0]  [1 0]  [1 0]  [1 1]  [1 0]  [1 0]  [1 1]

   [1 0]  [0 1]  [0 1]  [1 0]  [1 1]  [0 1]  [1 1]

   [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [1 1]

   [0 1]  [0 1]  [1 0]

MATHEMATICA

T[n_, k_] := With[{m = n k}, Sum[Binomial[Binomial[j, n] + k - 1, k] Sum[ (-1)^(i - j) Binomial[i, j], {i, j, m}], {j, 0, m}]];

Table[T[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Apr 10 2020, from PARI *)

PROG

(PARI) T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j, n)+k-1, k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}

CROSSREFS

Rows n=1..3 are A000012, A121316, A136246.

Columns k=0..3 are A000012, A000012, A226994, A137220.

The version with nonnegative integer entries is A331315.

Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:

All solutions: A262809 (all), A331567 (distinct rows).

Up to row permutation: A188392, A188445, A331126, A331039.

Up to column permutation: this sequence, A331571, A331277, A331569.

Nonisomorphic: A331461, A331510, A331508, A331509.

Cf. A331638.

Sequence in context: A303697 A202019 A295685 * A141471 A331572 A127080

Adjacent sequences:  A330939 A330940 A330941 * A330943 A330944 A330945

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Jan 13 2020

STATUS

approved

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Last modified June 1 09:27 EDT 2020. Contains 334759 sequences. (Running on oeis4.)