  A Skolem sequence is a sequence a_1,a_2,...,a_2n (where a_i \in A = {1,...,n
}), each a_i occurs exactly twice in the sequence and the two occurrences are
exactly a_i positions apart. A set A that can be used to construct Skolem
sequences is called a Skolem set. The problem of deciding which sets of the
form A = {1,...,n} are Skolem sets was solved by Thoralf Skolem in the late
1950's. We study the natural generalization where A is allowed to be any set of
n positive integers. We give necessary conditions for the existence of Skolem
sets of this generalized form. We conjecture these necessary conditions to be
sufficient, and give computational evidence in favor of our conjecture. We
investigate special cases of the conjecture and prove that the conjecture hold
for some of them. We also study enumerative questions and show that this
problem has strong connections with problems related to permutation
displacements.
