The intersection spectrum of hooked Skolem sequences and applications

A hooked Skolem sequence of order n is a sequence hS"n=(s"1,s"2,...,s"2"n"+"1) of 2n+1 integers containing each of the integers 1,2,...,n exactly twice, such that two occurrences of the integer j@?{1,2,...,n} are separated by exactly j-1 integers, and s"2"n=0. We prove that the necessary conditions are sufficient for the existence of two hooked Skolem sequences of order n with 0,1,2,...,n-3 and n pairs in the same positions. Further, we apply this result to the fine structure of cyclic three-fold triple systems and cyclic four-fold triple systems for v=13,19(mod24). Then, we extend these results to the fine structure of cyclic directed triple systems and cyclic Mendelsohn triple systems.