Reduced words and plane partitions. (English) Zbl 0882.05010

Fomin, Sergey; Kirillov, Anatol N.

J. Algebr. Comb. 6, No. 4, 311-319 (1997).

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

Keywords:

Young tableaux; Ferrers shape; reduced words; identity; Stanley’s formula; Macdonald’s formula; Schubert polynomials; enumeration of plane partitions; permutations; shapes; \(q\)-analogues

Citations:

Zbl 0579.05001

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