An n×n array with entries in [n] such that each integer appears exactly once in every row and every column is called a Latin square of order n. Two Latin squares L and L' are said to be orthogonal if, for all x,y∈[n], there is a unique pair (i,j) such that L(i,j) = x and L'(i,j) = y; k Latin squares are mutually orthogonal if any two of them are orthogonal.

After the question of existence of a combinatorial structure satisfying given properties, a natural and important problem is to determine how many such objects there are. In this talk, we will consider some counting questions related to (mutually) orthogonal Latin squares. We will present an upper bound on the number of ways to extend a set of k mutually orthogonal Latin squares to a set of k+1 mutually orthogonal Latin squares and discuss some applications, comparing the resulting bounds to previously known lower and upper bounds.
This talk is based on joint work with Shagnik Das and Tibor Szabó.