On Degrees in the Hasse Diagram of the Strong Bruhat Order

Details On Degrees in the Hasse Diagram of the Strong Bruhat Order On Degrees in the Hasse Diagram of the Strong Bruhat Order

2005-05-02

arxiv.org/abs/math/0505020v2

Mathematics

Combinatorics

14 pages, minor corrections; to appear in S\'em. Lothar. Combin

Scientific paper

For a permutation $\pi$ in the symmetric group $S_n$ let the {\it total degree} be its valency in the Hasse diagram of the strong Bruhat order on $S_n$, and let the {\it down degree} be the number of permutations which are covered by $\pi$ 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 Tur\'an from extremal graph theory.

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