Asymptotic enumeration of Latin rectangles

Abstract

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 K n, n . We prove that the number of k × n Latin rectangles is asymptotically (n!)( n(n−1)⋯(n−k+1) n k n(1− k n ) − n 2 e − k 2 as n → ∞ with k = o(n 6 7 ) . This improves substantially on previous work by Erdős 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.