Latin rectangle

In combinatorial mathematics, a Latin rectangle is an r × n matrix (where r ≤ n), using n symbols, usually the numbers 1, 2, 3, ..., n or 0, 1, ..., n − 1 as its entries, with no number occurring more than once in any row or column. An n × n Latin rectangle is called a Latin square. 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. An example of a 3 × 5 Latin rectangle is: {| class="wikitable" |- |0|| 1|| 2|| 3|| 4 |- |3|| 4 ||0 ||1|| 2 |- |4|| 0|| 3|| 2|| 1 |} A Latin rectangle is called normalized (or reduced) if its first row is in natural order and so is its first column. The example above is not normalized. Let L(k, n) denote the number of normalized k × n Latin rectangles. Then the total number of k × n Latin rectangles is A 2 × n Latin rectangle corresponds to a permutation with no fixed points. Such permutations have been called discordant permutations. An enumeration of permutations discordant with a given permutation is the famous problème des rencontres. The enumeration of permutations discordant with two permutations, one of which is a simple cyclic shift of the other, is known as the reduced problème des ménages. The number of normalized Latin rectangles, L(k, n), of small sizes is given by {| class="wikitable" style="text-align:center;" |-