On the Number of Maximum Inner Distance Latin Squares

The inner distance of a Latin square was defined by myself and six others during an REU in the Summer of 2020 at Moravian College. Since then, I have been curious about its possible connections to other combinatorial mathematics. The inner distance of a matrix is the minimum value of the distance between entries in adjacent cells, where our distance metric is distance modulo $n$. Intuitively, one expects that most Latin squares have inner distance 1, for example there probably exists a pair of adjacent cells with consecutive integers. And very few should have \textit{maximum} inner distance; the maximum inner distance was found by construction for all $n\geq 3$ to be exactly $\left \lfloor\frac{n-1}{2}\right \rfloor$. In this paper we also establish existence for all smaller inner distances. Much of our introductory work is showcased in \cite{inner distance}, with a primary focus on determining the maximum inner distance for Latin squares and Sudoku Latin squares. In this paper, we focus on enumerating maximum inner distance squares, and calculated the exact number for any $n$.