On Degrees in the Hasse Diagram of the Strong Bruhat Order

For a permutation π in the symmetric group Sn let the total degree be its valency in the Hasse diagram of the strong Bruhat order on Sn, and let the down degree be the number of permutations which are covered by π in the strong Bruhat order. The maxima of the total degree and the down degree and their values at a random permutation are computed. Proofs involve variants of a classical theorem of Turán from extremal graph theory. 1 The Down, Up and Total Degrees Definition 1.1 For a permutation π ∈ Sn let the down degree d−(π) be the number of permutations in Sn which are covered by π in the strong Bruhat order. Let the up degree d+(π) be the number of permutations which cover π in this order. The total degree of π is the sum d(π) := d−(π) + d+(π), i.e., the valency of π in the Hasse diagram of the strong Bruhat order. Department of Mathematics and Statistics, Bar-Ilan University, Ramat-Gan 52900, Israel. Email: [email protected] Department of Mathematics and Statistics, Bar-Ilan University, Ramat-Gan 52900, Israel. Email: [email protected] Research of both authors was supported in part by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities, and by the EC’s IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272.