The many formulae for the number of Latin rectangles

Abstract

A k×n Latin rectangle L is a k×n array, with symbols from a set of cardinality n, such that each row and each column contains only distinct symbols. If k = n then L is a Latin square. Let Lk,n be the number of k×n Latin rectangles. We survey (a) the many combinatorial objects equivalent to Latin squares, (b) the known bounds on Lk,n and approximations for Ln, (c) congruences satisfied by Lk,n and (d) the many published formulae for Lk,n and related numbers. We also describe in detail the method of Sade in finding L7,7, an important milestone in the enumeration of Latin squares, but which was privately published in French. Doyle's formula for Lk,n is given in a closed form and is used to compute previously unpublished values of L4,n, L5,n and L6,n. We reproduce the three formulae for Lk,n by Fu that were published in Chinese. We give a formula for Lk,n that contains, as special cases, formulae of (a) Fu, (b) Shao and Wei and (c) McKay and Wanless. We also introduce a new equation for Lk,n whose complexity lies in computing subgraphs of the rook's graph.