The Asymptotic Number of Latin Rectangles

1. Introduction. The problem of enumerating n by k Latin rectangles was solved formally by MacMahon [4] using his operational methods. For k = 3, more explicit solutions have been given in [1], [2], [3], and [5]. Wile further exact enumeration seems difficult, it is an easy heuristic conjecture that the number of n by k Latin rectangles is asymptotic to (-n!)'cexp (-),CY,). Because of an error, Jacob [2] was led to deny this conjecture for k = 3 ; but Kerawala [3] rectified the error and then verified the conjecture to a high degree of approximation. The first proof for k = 3 • appears to have been given by Riordan [5]. In this paper we shall prove the conjecture not only for k fixed (as It-> c) but for k < (loon) As indicated below, a considerably shorter proof could be given for the former case. The additional detail is perhaps justified by (1) the interest attached to an approach to Latin squares (k = n), (2) the emergence of further terms of an asymptotic series (4), (3) the fact that (log n) 3/ 1 appears to be a "natural boundary" of the method. (We believe however that the actual break occurs at k = n 1 /3 .) 2. Notation. An n by k Latin rectangle L is an array of n rows and k columns, with the integers 1, •. n in each row and all distinct integers in each column. Let N be the number of ways of adding a (k + 1)-st row to L so as to make the augmented array a Latin rectangle. We use the sieve method (method of inclusion and exclusion) to obtain an expression for X. From n !, the total number of possible choices for the (k + 1)-st row, we take away those having a clash with L in a given column-summed over all choices of that column, then reinstate those having clashes in two given columns, etc. The result can be written where A, is the number of ways of choosing r distinct integers in L, no two in the same column. In particular A, = 1, A 1 = nk. To estimate the higher values of A r we apply the sieve method again. The total number of ways of * Received November 30, 1945 .