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A331315
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Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n and columns in nonincreasing lexicographic order.
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14
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1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 14, 4, 1, 1, 8, 150, 128, 8, 1, 1, 16, 2210, 10848, 1288, 16, 1, 1, 32, 41642, 1796408, 933448, 13472, 32, 1, 1, 64, 956878, 491544512, 1852183128, 85862144, 143840, 64, 1, 1, 128, 25955630, 200901557728, 7805700498776, 2098614254048, 8206774496, 1556480, 128, 1
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OFFSET
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0,8
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COMMENTS
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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 n-uniform k-block multisets of multisets.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..1325
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FORMULA
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A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,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))*A316674(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331278(n, j).
A(n, k) = A011782(n) * A330942(n, k) for k > 0.
A317583(n) = Sum_{d|n} A(n/d, d).
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EXAMPLE
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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 2 14 150 2210 41642 ...
3 | 1 4 128 10848 1796408 491544512 ...
4 | 1 8 1288 933448 1852183128 7805700498776 ...
5 | 1 16 13472 85862144 2098614254048 140102945876710912 ...
6 | 1 32 143840 8206774496 2516804131997152 ...
...
The A(2,2) = 14 matrices are:
[1 0] [1 0] [1 0] [2 0] [1 1] [1 0] [1 0]
[1 0] [0 1] [0 1] [0 1] [1 0] [1 1] [1 0]
[0 1] [1 0] [0 1] [0 1] [0 1] [0 1] [0 2]
[0 1] [0 1] [1 0]
.
[1 0] [1 0] [2 1] [2 0] [1 1] [1 0] [2 2]
[0 2] [0 1] [0 1] [0 2] [1 1] [1 2]
[1 0] [1 1]
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PROG
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(PARI) T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1, n)+k-1, k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}
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CROSSREFS
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Rows n=1..2 are A000012, A121227.
Columns k=0..2 are A000012, A011782, A331397.
The version with binary entries is A330942.
The version with distinct columns is A331278.
Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
All solutions: A316674 (all), A331568 (distinct rows).
Up to row permutation: A219727, A219585, A331161, A331160.
Up to column permutation: this sequence, A331572, A331278, A331570.
Nonisomorphic: A331485.
Cf. A317583.
Sequence in context: A298261 A341991 A152937 * A360936 A361014 A064552
Adjacent sequences: A331312 A331313 A331314 * A331316 A331317 A331318
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KEYWORD
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nonn,tabl
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AUTHOR
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Andrew Howroyd, Jan 13 2020
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STATUS
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approved
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