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- W1485706475 abstract "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 $$sumlimits_{(a_1 ,a_2 , ldots ) in Red(w_0 )} {(x + a_1 )(x + a_2 )} cdots = left( {_2^n } right)!prodlimits_{1 leqslant i < j leqslant n} {frac{{2x + i + j - 1}}{{i + j - 1}}} ,$$ which generalizes Stanley's [20] formula forthe cardinality of Red(w 0), and Macdonald's [11] formula $$sum {a_1 a_2 cdots = (_2^n )} !$$ .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." @default.
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- W1485706475 doi "https://doi.org/10.1023/a:1008694825493" @default.
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