The Many Formulae for the Number of Latin Rectangles

Abstract

A $k \times n$ Latin rectangle $L$ is a $k \times n$ 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 $L_{k,n}$ be the number of $k \times n$ Latin rectangles. We survey (a) the many combinatorial objects equivalent to Latin squares, (b) the known bounds on $L_{k,n}$ and approximations for $L_n$, (c) congruences satisfied by $L_{k,n}$ and (d) the many published formulae for $L_{k,n}$ and related numbers. We also describe in detail the method of Sade in finding $L_{7,7}$, an important milestone in the enumeration of Latin squares, but which was privately published in French. Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$. We reproduce the three formulae for $L_{k,n}$ by Fu that were published in Chinese. We give a formula for $L_{k,n}$ 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 $L_{k,n}$ whose complexity lies in computing subgraphs of the rook's graph.

Eigenvalues and Eigenvectors of Latin Squares in Max-Plus Algebra

“…The study of Latin square has long tradition in combinatorics [1], for example the enumeration of Latin squares. The method or formula to enumerate the number of Latin squares can be found in [3,12,13]. An example of Latin square of order 4 is shown in below Example 1…”

Alternating Sign Matrices and Polynomiography

“…In contrast, the number of Latin Squares is much larger. Even for n = 6 their number is already large, 812851200, and for n = 7; 8; 9 their number is respectively (see [13,19] or Wikipedia on Latin Squares), 61479419904000 ≈ 6.1 × 10 13 , 108776032459082956800 ≈ 1.08 × 10 20 , 5524751496156892842531225600 ≈ 5.5 × 10 28 .…”

Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

“…The problem of counting r × s Latin rectangles based on n symbols is a classical problem in combinatorial design theory that is currently solved for r, s, n ≤ 11 (see [35] and the references therein). Their distribution into isotopism, isomorphism and main classes is only known for Latin squares of order n ≤ 11 [23,28,29].…”