On the Number of Latin Squares

Abstract

We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order 11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order n is divisible by f ! where f is a particular integer close to 1 2 n, (3) provide a formula for the number of Latin squares in terms of permanents of (+1, −1)-matrices, (4) find the extremal values for the number of 1-factorisations of k-regular bipartite graphs on 2n vertices whenever 1 ≤ k ≤ n ≤ 11, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases.

The theory and application of latin bitrades: A survey

“…However, latin squares, considered in their most general sense, arguably behave combinatorially rather than algebraically. For example, the number of latin squares of order n grows superexponentially in n, and most latin squares have a trivial autotopism group ( [55]). The focus of this survey is combinatorial.…”

“…For example, there are exactly twelve latin squares of order 3: So to generate a random latin square of order 3, it suffices to index the above set of squares in a list of size 12, then select, at random, an integer from 1 to 12. However, M c K a y and W a n l e s s [55] recently showed that the number of latin squares of order 11 is: 776966836171770144107444346734230682311065600000.…”

There are 1,132,835,421,602,062,347 nonisomorphic one‐factorizations of K14

“…[15]). In terms of running time, the entire computation (including the correctness checks described in what follows) took about 13 days on a Linux PC with a 3.66-GHz Intel Xeon CPU and 32 GB of main memory.…”

On the completability of incomplete orthogonal Latin rectangles

“…We call a Latin rectangle normalized if values 1, ..., n occur in the first row in natural order. Counting Latin rectangles is a topic broadly studied in combinatorics; some examples listed in chronological order are [16], [4], [7], [13] and [18]. Definition 1 Two m-row Latin rectangles of order n, with m < n, form an orthogonal pair (OLR) if and only if when superimposed each of the n 2 ordered pairs of values (1, 1), (1,2), ..., (n, n) appears at most once.…”