The Asymptotic Number of Latin Rectangles

Abstract

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

Locations

• American Journal of Mathematics - View
• CiteSeer X (The Pennsylvania State University) - View - PDF