The existence of circular Florentine arrays

Abstract

A κ × n circular Florentine array is an array of n distinct symbols in gk circular rows such that 1. (1) each row contains every symbol exactly once, and 2. (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 b 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.