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Reduced words and plane partitions. (English) Zbl 0882.05010
Summary: Let $$w_0$$ be the element of maximal length in the symmetric group $$S_n$$, and let $$\text{Red}(w_0)$$ be the set of all reduced words for $$w_0$$. We prove the identity $\sum_{(a_1,a_2,\dots)\in\text{Red}(w_0)}(x+ a_1)(x+ a_2)\cdots= {n\choose 2}! \prod_{1\leq i<j\leq n} {2x+ i+ j-1\over i+j-1},\tag{$$*$$}$ which generalizes Stanley’s formula for the cardinality of $$\text{Red}(w_0)$$, and Macdonald’s formula $$\sum a_1a_2\cdots= \left(\begin{smallmatrix} n\\ 2\end{smallmatrix}\right)!$$. Our approach uses an observation, based on a result by M. L. Wachs [J. Comb. Theory, Ser. A 40, 276-289 (1985; Zbl 0579.05001)], 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 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.

MSC:
 05A17 Combinatorial aspects of partitions of integers 05A15 Exact enumeration problems, generating functions 05E05 Symmetric functions and generalizations 05E15 Combinatorial aspects of groups and algebras (MSC2010) 14M15 Grassmannians, Schubert varieties, flag manifolds 20C30 Representations of finite symmetric groups 05E10 Combinatorial aspects of representation theory
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