The fine structures of three pairwise distinct Latin squares

Let L-k = (a(ij)((k)))(v x v) k = 1, 2, 3 be Latin squares on symbol set S containing v elements. The fine structure of three Latin squares L-1, L-2, L-3 is defined to be (t, s) if s = v(2) - c(1) and t = c(2), where c(1) is the number of cells (i, j) such that vertical bar{(a(ij)((1)), a(ij)((3))}vertical bar = I for I = 1, 2. Denote by Fin*(v) the set of all integer pairs (t, s) for which there exist three pairwise distinct Latin squares of order v on the same set having fine structure (t, s). We determine the set Fin*(v) for any positive integer v >= 8. (C) 2009 Elsevier B.V. All rights reserved.