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%I #21 Apr 11 2020 22:41:47
%S 1,1,2,2,4,2,7,2,10,7,12,2,38,2,21,46,72,2,162,2,420,415,64,2,4987,
%T 1858,110,9336,45456,2,136018,2,1014658,406578,308,3996977,34937078,2,
%U 502,28010167,1530292965,2,508164038,2,54902992348,51712929897,1269,2,3217847072904,8597641914,9168720349613
%N Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n.
%C 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.
%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%C 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, for some d|n.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Magic_square">Magic square</a>
%F a(p) = 2 for p prime corresponding to the 1 X 1 square [p] and the permutation matrices of size p X p with partition (1...10...0). - _Chai Wah Wu_, Jan 16 2019
%F a(n) = Sum_{d|n} A333733(d,n/d) for n > 0. - _Andrew Howroyd_, Apr 11 2020
%e Non-isomorphic representatives of the a(2) = 2 through a(6) = 7 multiset partitions:
%e {{11}} {{111}} {{1111}} {{11111}} {{111111}}
%e {{1}{2}} {{1}{2}{3}} {{11}{22}} {{1}{2}{3}{4}{5}} {{111}{222}}
%e {{12}{12}} {{112}{122}}
%e {{1}{2}{3}{4}} {{11}{22}{33}}
%e {{11}{23}{23}}
%e {{12}{13}{23}}
%e {{1}{2}{3}{4}{5}{6}}
%e Inequivalent representatives of the a(6) = 7 matrices:
%e [6]
%e .
%e [3 0] [2 1]
%e [0 3] [1 2]
%e .
%e [2 0 0] [2 0 0] [1 1 0]
%e [0 2 0] [0 1 1] [1 0 1]
%e [0 0 2] [0 1 1] [0 1 1]
%e .
%e [1 0 0 0 0 0]
%e [0 1 0 0 0 0]
%e [0 0 1 0 0 0]
%e [0 0 0 1 0 0]
%e [0 0 0 0 1 0]
%e [0 0 0 0 0 1]
%e Inequivalent representatives of the a(9) = 7 matrices:
%e [9]
%e .
%e [3 0 0] [3 0 0] [2 1 0] [2 1 0] [1 1 1]
%e [0 3 0] [0 2 1] [1 1 1] [1 0 2] [1 1 1]
%e [0 0 3] [0 1 2] [0 1 2] [0 2 1] [1 1 1]
%e .
%e [1 0 0 0 0 0 0 0 0]
%e [0 1 0 0 0 0 0 0 0]
%e [0 0 1 0 0 0 0 0 0]
%e [0 0 0 1 0 0 0 0 0]
%e [0 0 0 0 1 0 0 0 0]
%e [0 0 0 0 0 1 0 0 0]
%e [0 0 0 0 0 0 1 0 0]
%e [0 0 0 0 0 0 0 1 0]
%e [0 0 0 0 0 0 0 0 1]
%Y Cf. A006052, A007716, A057150, A120732, A271103, A319056, A319616.
%Y Cf. A321717, A321718, A321719, A321722, A321724, A333733.
%K nonn
%O 0,3
%A _Gus Wiseman_, Nov 18 2018
%E a(11)-a(13) from _Chai Wah Wu_, Jan 16 2019
%E a(14)-a(15) from _Chai Wah Wu_, Jan 20 2019
%E Terms a(16) and beyond from _Andrew Howroyd_, Apr 11 2020
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