ABSTRACT
Fulton introduced the essential set of a permutation, together with a rank function. In this paper we study some combinatorial aspects of the essential set: We present an algorithm that retrieves a permutation from its ranked essential set. Thereby we can characterize the class of ranked sets that arise as ranked essential sets of permutations, answering a question of Fulton. Several classes of permutations are characterized in terms of their essential set. Results in higher dimensions are discussed.