The maximum number of atoms of Bruhat intervals in the symmetric groups

We prove that the maximum number of (co)atoms of Bruhat intervals of the length n − 1 in the symmetric group S n is ⌊ n 2 / 4 ⌋ . We show how to construct such an interval, explicitly making use of the subexpression property among bigrassmannian permutations together with the result by Adin–Roichman that the maximum of the down degree (the number of elements covered by a given permutation) in S n is ⌊ n 2 / 4 ⌋ .