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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.

%I A330942 #29 Apr 10 2020 08:16:41
%S A330942 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,
%T A330942 32,20821,449102,116681,842,1,1,1,64,478439,122886128,231522891,
%U A330942 5366384,4495,1,1,1,128,12977815,50225389432,975712562347,131163390878,256461703,24320,1,1
%N 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.
%C A330942 The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
%C A330942 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.
%H A330942 Andrew Howroyd, <a href="/A330942/b330942.txt">Table of n, a(n) for n = 0..1325</a>
%F A330942 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)).
%F A330942 A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A262809(n, j)/k!.
%F A330942 A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331277(n, j).
%F A330942 A331638(n) = Sum_{d|n} A(n/d, d).
%e A330942 Array begins:
%e A330942 ============================================================
%e A330942 n\k | 0 1    2         3              4                5
%e A330942 ----+-------------------------------------------------------
%e A330942   0 | 1 1    1         1              1                1 ...
%e A330942   1 | 1 1    2         4              8               16 ...
%e A330942   2 | 1 1    7        75           1105            20821 ...
%e A330942   3 | 1 1   32      2712         449102        122886128 ...
%e A330942   4 | 1 1  161    116681      231522891     975712562347 ...
%e A330942   5 | 1 1  842   5366384   131163390878 8756434117294432 ...
%e A330942   6 | 1 1 4495 256461703 78650129124911 ...
%e A330942   ...
%e A330942 The A(2,2) = 7 matrices are:
%e A330942    [1 0]  [1 0]  [1 0]  [1 1]  [1 0]  [1 0]  [1 1]
%e A330942    [1 0]  [0 1]  [0 1]  [1 0]  [1 1]  [0 1]  [1 1]
%e A330942    [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [1 1]
%e A330942    [0 1]  [0 1]  [1 0]
%t A330942 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}]];
%t A330942 Table[T[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Apr 10 2020, from PARI *)
%o A330942 (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)))}
%Y A330942 Rows n=1..3 are A000012, A121316, A136246.
%Y A330942 Columns k=0..3 are A000012, A000012, A226994, A137220.
%Y A330942 The version with nonnegative integer entries is A331315.
%Y A330942 Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
%Y A330942 All solutions: A262809 (all), A331567 (distinct rows).
%Y A330942 Up to row permutation: A188392, A188445, A331126, A331039.
%Y A330942 Up to column permutation: this sequence, A331571, A331277, A331569.
%Y A330942 Nonisomorphic: A331461, A331510, A331508, A331509.
%Y A330942 Cf. A331638.
%K A330942 nonn,tabl
%O A330942 0,8
%A A330942 _Andrew Howroyd_, Jan 13 2020