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A balanced matrix is a binary matrix (entries are 0 or 1) such that each row and each column add up to the same value.

This definition works for non-square matrices, though the row sums will be different from the column sums (unless both are zero).

Square balanced matrices can be divided into classes B(n, k) which contain nxn matrices with k 1's in each row and column.

We can consider two balanced matrices of the same shape to be similar if one can be obtained from the other by a permuation of rows and columns. This includes flipping a matrix horizontally or vertically, but does not include diagonal flips.

Similarity of matrices induces a partition on B(n, k). Below are examples from the four types of matrices in B(6, 2).

Connection to Latin Squares

If we take an nxn Latin square and apply an arbitrary function mapping [n] to {0, 1} to each entry in the square, we obtain a balanced matrix.

We can define an undirected graph with vertices being permutations on [n] with two vertices being adjacent if and only if the two permutations are derangements of each other.


Cliques in this graph correspond to balanced matrices and a set of Latin rectangles and balanced matrices. Maximal cliques, which all contain n vertices, correspond to n! Latin squares each.