The intersection spectrum of hooked Skolem sequences and applications

A hooked Skolem sequence of order nn is a sequence hSn=(s1,s2,…,s2n+1)hSn=(s1,s2,…,s2n+1) of 2n+12n+1 integers containing each of the integers 1,2,…,n1,2,…,n exactly twice, such that two occurrences of the integer j∈{1,2,…,n}j∈{1,2,…,n} are separated by exactly j−1j−1 integers, and s2n=0s2n=0. We prove that the necessary conditions are sufficient for the existence of two hooked Skolem sequences of order nn with 0,1,2,…,n−30,1,2,…,n−3 and nn 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.