On the Number of Reduced Decompositions of Elements of Coxeter Groups

Let r( w) denote the number of reduced decompositions of the element w of a Coxeter group W Using the theory of symmetric functions, a formula is found for r( w) when W is the symmetric group S". For the element Wo E S" of longest length and certain other WE S", the formula for r( w) is particularly simple. For the hyperoctahedral group Bn some conjectures are made in analogy to the Sn case. The situation for other W remains unclear.