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There are 5 nonisomorphic 4 X 4 matrices with row and column sums 2:
[0 0 0 2] [0 0 0 2] [0 0 0 2] [0 0 1 1] [0 0 1 1]
[0 0 2 0] [0 0 2 0] [0 1 1 0] [0 0 1 1] [0 1 0 1]
[0 2 0 0] [1 1 0 0] [1 0 1 0] [1 1 0 0] [1 0 1 0]
[2 0 0 0] [1 1 0 0] [1 1 0 0] [1 1 0 0] [1 1 0 0]
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