The intersection spectrum of hooked Skolem sequences and applications
A hooked Skolem sequence of order n n is a sequence hSn=(s1,s2,,s2n+1) h S n = ( s 1 , s 2 ,  , s 2 n + 1 ) of 2n+1 2 n + 1 integers containing each of the integers 1,2,,n 1 , 2 ,  , n exactly twice, such that two occurrences of the integer j?{1,2,,n} j ? { 1 , 2 ,  , n } are separated by exactly j?1 j ? 1 integers, and s2n=0 s 2 n = 0 . We prove that the necessary conditions are sufficient for the existence of two hooked Skolem sequences of order n n with 0,1,2,,n?3 0 , 1 , 2 ,  , n ? 3 and n 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 ( mod 24 ) . Then, we extend these results to the fine structure of cyclic directed triple systems and cyclic Mendelsohn triple systems.
mendelsohn triple systems skolem and langford sequences combinatorial designs cyclic triple systems