Reduced Words and Plane Partitions

Let w0 be the element of maximal length in thesymmetric group Sn, and let Red(w0) bethe set of all reduced words for w0. We prove the identity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\limits_{(a_1 ,a_2 , \ldots ) \in Red(w_0 )} {(x + a_1 )(x + a_2 )} \cdots = \left( {_2^n } \right)!\prod\limits_{1 \leqslant i < j \leqslant n} {\frac{{2x + i + j - 1}}{{i + j - 1}}} ,$$ \end{document}which generalizes Stanley's [20] formula forthe cardinality of Red(w0), and Macdonald's [11] formula\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum {a_1 a_2 \cdots = (_2^n )} !$$ \end{document}.Our approach uses anobservation, based on a result by Wachs [21], that evaluation of certainspecializations of Schubert polynomials is essentially equivalent toenumeration of plane partitions whose parts are bounded from above. Thus,enumerative results for reduced words can be obtained from the correspondingstatements about plane partitions, and vice versa. In particular, identity(*) follows from Proctor's [14] formula for the number of planepartitions of a staircase shape, with bounded largest part.Similar results are obtained for other permutations and shapes;q-analogues are also given.