Reduced Words and Plane Partitions

Abstract

Let w0 be the element of maximal length in the symmetric group Sn, and let Red(w0) be the set of all reduced words for w0. We prove the identity (Equation Presented) which generalizes Stanley's [20] formula for the cardinality of Red(w0), and Macdonald's [11] formula ∑a1a2 ⋯ = (Equation Presented)!. Our approach uses an observation, based on a result by Wachs [21], that evaluation of certain specializations of Schubert polynomials is essentially equivalent to enumeration of plane partitions whose parts are bounded from above. Thus, enumerative results for reduced words can be obtained from the corresponding statements about plane partitions, and vice versa. In particular, identity (*) follows from Proctor's [14] formula for the number of plane partitions of a staircase shape, with bounded largest part. Similar results are obtained for other permutations and shapes; q-analogues are also given.