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A321724
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Irregular triangle read by rows where T(n,d) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d|n.
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7
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 1, 1, 5, 1, 1, 3, 7, 1, 1, 1, 1, 4, 9, 12, 11, 1, 1, 1, 1, 4, 15, 1, 1, 13, 31, 1, 1, 5, 43, 22, 1, 1, 1, 1, 5, 22, 103, 30, 1, 1, 1, 1, 6, 106, 264, 42, 1, 1, 30, 383, 1, 1, 6, 56, 1, 1, 1, 1, 7, 45, 321, 2804, 1731, 77, 1
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OFFSET
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1,7
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COMMENTS
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Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d.
A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1
1 1
1 1
1 2 1
1 1
1 2 3 1
1 1
1 3 5 1
1 5 1
1 3 7 1
Inequivalent representatives of the a(10,5) = 7 semi-magic squares (zeros not shown):
[2 ] [2 ] [2 ] [2 ] [2 ] [11 ] [11 ]
[ 2 ] [ 2 ] [ 2 ] [ 11 ] [ 11 ] [11 ] [1 1 ]
[ 2 ] [ 2 ] [ 11 ] [ 11 ] [ 1 1 ] [ 11 ] [ 1 1 ]
[ 2 ] [ 11] [ 1 1] [ 11] [ 1 1] [ 1 1] [ 1 1]
[ 2] [ 11] [ 11] [ 11] [ 11] [ 11] [ 11]
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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