AbstractBigrassmannian permutations are known as permutations which have preciselyone left descent and one right descent. They play an important role in the studyof Bruhat order. Fulton introduced the essential set of a permutation and studiedits combinatorics. As a consequence of his work, it turns out that the essential setof bigrassmannian permutations consists of precisely one element. In this article,we generalize this observation for essential sets of arbitrary permutations. Ourmain theorem says that there exists a bijection between bigrassmanian permutationsmaximal below a permutation and its essential set. For the proof, we make useof two equivalent characterizations of bigrassmannian permutations by Lascoux-Sch¨utzenberger and Reading...