Reduced words and plane partitions

Fomin, Sergey ; Kirillov, Anatol N.

Journal of Algebraic Combinatorics, Tome 6 (1997) no. 4, pp. 311-319.

Résumé

Summary: Let $w _{0}$ be the element of maximal length in thesymmetric group $S _{ n }$, and let $Red( w _{0})$ bethe set of all reduced words for $w _{0}$. We prove the identity å $_{( a $_1$ , a $_2$ , \frac{1}{4} ) \~I Red( w $_0$ )} ( x + a _{1} )( x + a _{2} )$ frac14 = $( _{2} ^{ n} )$! Õ $_{1 \leqslant i j \leqslant n} \frac2 x + i + j - 1 i + j - 1$ , sumlimits_(a_1 ,a_2 , $\ldots ) \in Red(w_0 )$ (x + a_1 )(x + a_2 ) $\cdots = \left( {_2^n } \right)$!prodlimits_$1 \leqslant $i j $\leqslant n$ frac2x + i + j - 1i + j - 1 , which generalizes Stanley's [20] formula forthe cardinality of $Red( w _{0})$, and Macdonald's [11] formula å $a _{1} a _{2}$ frac14 = $( _{2} ^{ n} ) ! \sum $a_1 a_2 $\cdots $= (_2^n ) ! .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.

Détail

Keywords: reduced word, plane partition, Schubert polynomial