Maximal degree in the Strong Bruhat Order of Bn

Given a permutation π ∈ Sn, let Γ−(π) be the graph on n vertices {1, . . . , n} where two vertices i < j are adjacent if π(i) > π(j) and there are no integers k, i < k < j, such that π(i) > π(k) > π(j). Let Γ(π) be the graph obtained by dropping the condition that π(i) > π(j), i.e. two vertices are adjacent if the rectangle [i, π(i)] × [j, π(j)] is empty. In the study of the strong order on permutation, Adin and Roichman introduced these graphs and computed their maximum number of edges. We generalize these results to the Weyl group of signed permutations Bn, working with graphs on vertices {−n, . . . , n} \ {0}, using new variants of a classical theorem of Turán.