ASYMPTOTIC ENUMERATION OF LATIN RECTANGLES

A k × n Latin rectangle is a k × n matrix with entries from {1,2, ..., n} such that no entry occurs more than once in any row or column. Equivalently, it is an ordered set of k disjoint perfect matchings of Kn,n. We prove that the number of k × n Latin rectangles is asymptotically (n!)( n(n-1)⋯(n-k+1) nkn(1- k n)- n 2e- k 2 as n → ∞ with k = o(n 6 7). This improves substantially on previous work by Erdo{combining double acute accent}s and Kaplansky, Yamamoto, and Stein. We also derive an asymptotic approximation to the generalised ménage numbers, and establish a number of results on entries in random Latin rectangles. © 1990.