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A331278
Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.
6
1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 12, 4, 1, 0, 1, 124, 124, 8, 1, 0, 1, 1800, 10596, 1280, 16, 1, 0, 1, 33648, 1764244, 930880, 13456, 32, 1, 0, 1, 769336, 484423460, 1849386640, 85835216, 143808, 64, 1, 0, 1, 20796960, 198461691404, 7798297361808, 2098356708016, 8206486848, 1556416, 128, 1
OFFSET
0,9
COMMENTS
The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
A(n,k) is the number of n-uniform k-block sets of multisets.
LINKS
FORMULA
A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A316674(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331315(n, j).
A(n, k) = A011782(n) * A331277(n, k) for k > 0.
A331395(n) = Sum_{d|n} A(n/d, d).
EXAMPLE
Array begins:
====================================================================
n\k | 0 1 2 3 4 5
----+---------------------------------------------------------------
0 | 1 1 0 0 0 0 ...
1 | 1 1 1 1 1 1 ...
2 | 1 2 12 124 1800 33648 ...
3 | 1 4 124 10596 1764244 484423460 ...
4 | 1 8 1280 930880 1849386640 7798297361808 ...
5 | 1 16 13456 85835216 2098356708016 140094551934813712 ...
6 | 1 32 143808 8206486848 2516779512105152 ...
...
The A(2,2) matrices are:
[1 0] [1 0] [1 0] [2 0] [1 1] [1 0]
[1 0] [0 1] [0 1] [0 1] [1 0] [1 1]
[0 1] [1 0] [0 1] [0 1] [0 1] [0 1]
[0 1] [0 1] [1 0]
.
[1 0] [1 0] [1 0] [2 1] [2 0] [1 0]
[1 0] [0 2] [0 1] [0 1] [0 2] [1 2]
[0 2] [1 0] [1 1]
PROG
(PARI) T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1, n), k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}
CROSSREFS
Rows n=1..2 are A000012, A173219.
Columns k=0..2 are A000012, A011782, A331396.
The version with binary entries is A331277.
The version with not necessarily distinct columns is A331315.
Cf. A316674 (unrestricted version), A330942, A331395.
Sequence in context: A326476 A247864 A370398 * A266632 A062154 A359619
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jan 13 2020
STATUS
approved