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A323349
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Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.
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9
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1, 1, 3, 3, 6, 3, 11, 3, 12, 6, 13, 3, 52, 3, 15, 30, 57, 3, 156, 3, 238, 129, 19, 3, 2221, 6, 21, 415, 3114, 3, 14921, 3, 12853, 1044, 25, 6219, 164743, 3, 27, 2220, 851476, 3, 954088, 3, 434106, 3326714, 31, 3, 24648724, 6, 22309800, 7269, 2737618, 3, 69823653
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OFFSET
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0,3
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COMMENTS
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Also the number of non-normal semi-magic rectangles summing to n with no zeros.
Matrices must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 3 for p prime, since the only allowable matrices must be of size 1 X 1, 1 X p or p X 1 with only one way to fill in the entries for each matrix size. Similarly, a(p^2) = 6 with additional allowable matrices of sizes 1 X p^2, p^2 X 1 and p X p, again with only one way to fill in the entries for each size. - Chai Wah Wu, Jan 13 2019
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LINKS
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FORMULA
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a(p) = 3 and a(p^2) = 6 for p prime (see comment). - Chai Wah Wu, Jan 13 2019
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EXAMPLE
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The a(6) = 11 matrices:
[6] [3 3] [2 2 2] [1 1 1 1 1 1]
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[3] [1 2] [2 1] [1 1 1]
[3] [2 1] [1 2] [1 1 1]
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[2] [1 1]
[2] [1 1]
[2] [1 1]
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[1]
[1]
[1]
[1]
[1]
[1]
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MATHEMATICA
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Table[Length[Select[Join@@Table[Partition[cmp, d], {cmp, Join@@Permutations/@IntegerPartitions[n]}, {d, Divisors[Length[cmp]]}], And[SameQ@@Total/@#, SameQ@@Total/@Transpose[#]]&]], {n, 10}]
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CROSSREFS
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Cf. A000219, A006052, A120733, A305551, A319056, A321719, A323295, A323300, A323302, A323306, A323347, A323349.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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