A combinatorial derivation of the Poincare polynomials of the finite irreducible Coxeter groups

(1) The Poincare polynomials of the finite irreducible Coxeter groups are derived by an elementary combinatorial method avoiding the use of Lie theory and invariant theory. (2) Non-recursive methods for the computation of 'standard reduced words' for (signed) permutations are described. The algebraic basis for both (1) and (2) is a simple partition property of the weak Bruhat order of Coxeter groups into isomorphic parts. (C) 2001 Elsevier Science B.V. All rights reserved.