A Primer on Latin Squares With Some Research Objectives

Latin Squares

A Latin square of order n is a square array that contains n different elements all occurring n times but with none occurring more than once in the same row or column. That’s a little wordy, it’s much easier to see an example of a Latin square. Below is a Latin square of order 4.

Normalized Latin Squares

Below are two more examples of Latin squares, the first of order 3 and the second of order 5. Notice that, unlike the square above, the first row and column of the squares are in ascending numerical order.

“Reduced” Latin Squares

Although this terminology seems to be non-standard, I was taught that a reduced Latin square is one in which the first row is given by 1, 2, …, n. That is, a reduced Latin square is almost a normalized Latin square but the first column need not be in ascending numerical order. The first order 4 square presented in this post is a reduced Latin square but isn’t normalized.

Isotopy Classes

Two Latin squares belong to the same isotopy class if one square can be obtained from the other by permuting the rows, columns, and symbols of the other square. More simply, given the three operations of swapping rows, swapping columns, and swapping symbols (e.g. all 3’s become 1’s and all 1’s become 3’s) if, when applying these three operations, we end up with a new Latin square than the newly created Latin square is in the same isotopy class as the first. Two Latin squares are said to be isotopic or to have isotopy class equivalence if they belong to the same isotopy class.

(Mutually) Orthogonal Latin Squares

Latin square orthogonality was the focal point of my undergraduate thesis. Two distinct n x n Latin squares, A = (a_ij) and B = (b_ij), are said to be orthogonal if the ordered pairs (a_ij, b_ij) are all distinct, here a_ij and b_ij are the elements of the Latin squares at row i and column j. A set of Latin squares each orthogonal to one another is called mutually orthogonal, and the squares are mutually orthogonal Latin squares (MOLS). This property, like many of the others, is most easily understood via example. Below is a set of two order 4 Latin squares which are orthogonal.

Current Research

My undergraduate research consisted of studying Latin squares. Having never taken a combinatorics course, the “research” began as a self-study of these objects. Eventually, I was tasked with determining the orthogonality of Latin squares of orders 2–6. Interestingly, while looking at the isotopy classes that the orthogonal Latin squares “came from” it was determined that all of the orthogonal squares were also isotopic. Furthermore, the isotopy class representative of this isotopy class happened to be symmetric. Exploring this further it was shown to be true for order 5, trivially true for order 4 (all isotopy class representatives are symmetric), and vacuously true for order 6 (there are no orthogonal squares of order 6). Initially, the continuation of my research consisted of determining if this conjecture was true for higher orders, i.e. are orthogonal Latin squares isotopic to a symmetric Latin square. To continue to study this for higher-order Latin squares, efficient algorithms to handle large numbers of Latin squares needed to be created.

I am a professional software engineer and an amateur mathematician. My main interests are programming, machine learning, fluid dynamics, and a few others.

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