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
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�ois 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
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jan 13 2020
STATUS
approved