INTRODUCTION

In other combinatorics courses you will see the techniques, both simple and sophisticated, that are used by combinatorialists to solve problems. Only in the Graph Theory course do you see a combinatorial structure (i.e. the graph), one of the objects that combinatorialists try to solve problems about. There are many other structures besides graphs that combinatorialists examine and we shall see a number of them in this course. While it is possible to conduct this course as a kind of zoological survey - bring out the strange beasts, examine them and then put them back in their cages - we will try to do something a bit more substantial. In order to do this we will need a theme, a common thread to hold together the variety of objects that we will look at. The theme that I've selected is that of interconnectedness. Any worker in the field of combinatorics soon comes to the realization that below the surface the different areas of the subject are strongly connected and any particular problem can be viewed from a number of different perspectives. This wealth of approaches to a problem gives the field some of its particular charm. The way in which this theme will be realized is by using one structure as a tie line connected to all the structures we will examine. The thread for us will be Latin squares.

Before we begin our journey in the jungle of combinatorial structures, a few words about procedure are in order. Whenever we meet a new combinatorial object there are two fundamental questions that arise. The first is the question of existence. Often, the structures we meet will be defined in terms of certain parameters and we will want to know which values of these parameters produce the objects in question and which do not. Sometimes, the search for an answer to this question will lead us to construction problems. The second fundamental question is that of uniqueness. If we know that there exists an example of a structure for a certain choice of the parameters, the next question to ask is how many examples are there? One often finds an inverse relationship in the difficulty of answering these two questions, that is if the existence question is easy then the uniqueness question is often very difficult and vice-versa. Of course, there are many cases in which neither question is easily answered.

Another point to keep in mind on our journey: this jungle is far from tamed. We will constantly be brushing the frontiers of knowledge and many easily stated questions will not have known answers or even hints at how they may be answered.

It is now time to start our investigations. The first exhibit on your right is that of Latin squares....

I. LATIN SQUARES

A Latin square is an n x n square matrix whose entries consist of n symbols such that each symbol appears exactly once in each row and each column. It is typical to use as the symbols the integers from 1 ... n.

Some examples:

12
21
123
231
312
1234
2341
3412
4123
1234
2143
3412
4321

The first three examples all follow the same scheme; each row is determined by cyclically rotating the previous row to the right. It is not hard to see that this procedure will work for any size n to produce a Latin square, so the existence question is easily answered - there exists a Latin square of any order n. The last two examples show that you can have more than one Latin square of a given order. There is a natural question to ask at this point - How many Latin squares of a given order are there? - but before we examine this question, let's conduct a little experiment.

[To be done in class - approx. 3 min] Construct a Latin square of order 5 which is not the cyclic square.

Given the variety of answers, one wonders if they are all really different or is there some sense in which we can say that different looking squares are equivalent?

There are three operations that we can perform on a Latin square which will preserve the "Latinness" of the square. They are:

  1. Permute the rows
  2. Permute the columns
  3. Permute the symbols (i.e., rename the symbols without changing their relative positions).
If we can change one square into another by means of any or all of these operations, we say that the two squares are isotopic. Isotopy is an equivalence relation on the set of Latin squares of a given order (prove this - reflexive, symmetric and transitive - for homework). Thus it makes sense to call only nonisotopic squares different. Note that the two examples of order 4 squares are nonisotopic.

Given a square we can permute the columns so that the first row consists of 1 2 3 4 .... n in their natural order. After doing this we could permute the rows so that the first column is also 1 ... n in the natural order. The resulting square is of course isotopic to the original square and is a convenient representative of the isotopy class of this square. Such a square is said to be in standard form or reduced. (There will in general be more than one reduced square in an isotopy class. Find two reduced squares of order 4 in the same isotopy class - homework)

Now lets look at this list of order 5 squares. How many different (i.e., nonisotopic ) squares are there?

Let's back up a little. There is only one isotopy class of order 2 squares, and only one for order 3 squares. There are two classes for order four squares (although there are four reduced squares of this order). There are 2 classes of order 5 (with 56 reduced squares) and 22 classes of order 6. Other known results are:

Order # Isotopy Classes # Reduced Squares
6 22 9,408
7 563 16,942,080
8 1,676,257 535,281,401,856
9 ? 377,597,570,964,258,816

The question of enumerating and classifying Latin squares is not an easy one. Give some thought to how you might attempt to do this.

Latin squares have a long history. The concept probably originated with problems concerning the movement and disposition of pieces on a chess board. However, the earliest written reference concerned the problem of placing the 16 face cards of an ordinary playing deck in the form of a square so that no row, column or diagonal should contain more than one card of each suit and each rank. An enumeration by type of the solutions to this problem was published in 1723. A similar problem involving the arrangement of 36 officers of 6 different ranks and regiments in a square phalanx was proposed by Euler in 1779, but not until the beginning of the present century was it shown that no solution was possible. The Latin square concept certainly goes back further than these written documents. In his famous etching Melencolia, the 15th Century artist Albrecht Dürer portrays an order 4 magic square, a relative of Latin squares, in the background. Magic squares can also be found in the ancient chinese literature.

The systematic development of Latin squares started with Euler (1779) and was carried on by Cayley (1877-1890) who showed that the multiplication table of a group is an appropriately bordered special Latin square. In the 1930's the concept arose once again in the guise of multiplication tables when the theory of quasi-groups and loops began to be developed as a generalization of the group concept. Latin squares played an important role in the foundations of finite geometries, a subject which was also in development at this time. Also in the 1930's, a large application area for Latin squares was opened by R.A.Fisher who used them and other combinatorial structures in the design of statistical experiments.

Let us look at an example of the use of Latin squares in the design of a statistical experiment.

Cox [The Planning of Experiments, 1958] discusses an experiment in prosthodontics which compares seven treatments, which are commercial dentures of different materials and set at different angles. It is desirable to eliminate as much as possible of the variation due to differences between patients. Hence, each patient wears dentures of one type for a month, then dentures of another type for another month, and so on. After seven months, each patient has worn each type of denture.

In this experiment, it seems likely that the results in later months will be different from those in earlier months, and hence it is sensible to arrange that each treatment be used equally often in each time position. Thus, there are two types of variation, namely between-patient and between-time variation. The desire to balance out both types suggests the use of Latin squares.

We could design this experiment in the following way. Form a 7 x 7 square whose columns are labelled with the months and whose rows are labelled with the patients.

  Jan Feb Mar Apr May Jun Jul
Pat. A        
B       
C       
D       
E       
F       
G       

Now associate a digit from 1 - 7 with each of the treatments and fill in the cells of the square with (any) order 7 Latin square. The square is then the experimental design, specifying which treatment is given to which patient for each month. Since the design is a Latin square, each patient will get each treatment and each treatment will be used each month, thus balancing out the variations.

REFERENCES

Denes,J. & Keedwell,A.D., Latin Squares and Their Applications, 1974, Academic Press, New York. [Out of Print - but a second edition has come out]

This book is a compilation of the vast literature on Latin squares. The style is a bit erratic since many research papers were simply rewritten to put this book together; however, it is invaluable as a resource book and we will refer to it often.

Laywine, C. & Mullen, G., Discrete Mathematics Using Latin Squares, John Wiley & Sons, Inc., New York, 1998.

This book has just been published, and I haven't seen a copy of it yet, but the table of contents looks very promising.