The existence of circular Florentine arrays

A k × n circular Florentine array is an array of n distinct symbols in k circular rows such that (1) each row contains every symbol exactly once, and (2) for any pair of distinct symbols (a, b) and for any integer m from 1 to n - 1 there is at most one row in which 6 occurs m steps to the right of a. For each positive integer n = 2,3,4,..., define F_c(n) to be the maximum number such that an F_c(n) × n circular Florentine array exists. From the main construction of this paper for a set of mutually orthogonal Latin squares (MOLS) having an additional property, and from the known results on the existence/nonexistence of such MOLS obtained by others, it is now possible to reduce the gap between the upper and lower bounds on F_c(n) for infinitely many additional values of n not previously covered. This is summarized in the table for all odd n up to 81.