On the number of reduced decompositions of elements of Coxeter groups (Q1072636)

Let r(w) denote the number of reduced decompositions of the element w of a Coxeter group W. The author gives a formula for r(w) in terms of symmetric functions when W is the symmetric group \(S_ n\) (Weyl group of type A). This formula is quite explicit in many cases, e.g. if \(w_ 0\) is the element of maximal length in \(S_ n\), then \(r(w_ 0)\) is equal to the number of standard Young tableaux of the (staircase) shape (n-1,n- 2,...,1). When W is the hyperoctahedral group (Weyl group of type B) the author formulates some conjectures for r(w) in analogy to the \(S_ n\) case in terms of shifted standard tableaux. The situation for other Weyl groups remains unclear.