Latin rectangles and Latin squares may also be described as the optimal colorings of rook's graphs, or as optimal edge colorings of complete bipartite graphs.
[4] The number of normalized Latin rectangles, L(k, n), of small sizes is given by[6] When k = 1, that is, there is only one row, since the Latin rectangles are normalized there is no choice for what this row can be.
The table also shows that L(n − 1, n) = L(n, n), which follows since if only one row is missing, the missing entry in each column can be determined from the Latin square property and the rectangle can be uniquely extended to a Latin square.
A semi-Latin square is an n × n array, L, in which some positions are unoccupied and other positions are occupied by one of the integers {0, 1, ..., n − 1}, such that, if an integer k occurs in L, then it occurs n times and no two k's belong to the same row or column.
[8] One way to prove this is to observe that a semi-Latin square of order n and index m is equivalent to an m × n Latin rectangle.