1,3

The Bruhat graph on S_n is the directed graph with an edge connecting v to w whenever v and w differ by a transposition and w has more inversions than v. A maximal path in the Bruhat graph on S_n is one which goes from the identity element to the longest permutation [n,n-1,..., 2,1] written in one-line notation. Note, the Bruhat graph has more edges than the Hasse diagram for the Bruhat order. For example in S_3, [123] is connected to [321] in the Bruhat graph because they differ by a single transposition.

Anders Bjorner and Francesco Brenti, "Combinatorics of Coxeter Groups". Graduate Texts in Mathematics, 231. Springer, New York, 2005.

Francesco Brenti, "Lattice paths and Kazhdan-Lusztig polynomials" J. Amer. Math. Soc., 11 (1998), 229-259.

James Carrell, "The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties". Proceedings of Symposia in Pure Math.}, 56 (1994), 53--61.

Cf. A061710

Sequence in context: A027682 A006413 A055316 this_sequence A004784 A144796 A119657

Adjacent sequences: A165205 A165206 A165207 this_sequence A165209 A165210 A165211

hard,nonn

Sara C. Billey (billey(AT)uw.edu), Sep 07 2009