# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Billey, S. C.; Jockusch, W.; Stanley, R. P., Some combinatorial properties of Schubert polynomials, J. Alg. Combin., 2, 345-374, (1993) · Zbl 0790.05093  Edelman, P.; Greene, C., Balanced tableaux, Advances in Math., 63, 42-99, (1987) · Zbl 0616.05005  Fomin, S.; Stanley, R. P., Schubert polynomials and the nilCoxeter algebra, Advances in Math., 103, 196-207, (1994) · Zbl 0809.05091  Fomin, S.; Kirillov, A. N., The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math., 153, 123-143, (1996) · Zbl 0852.05078  Fomin, S.; Greene, C.; Reiner, V.; Shimozono, M., Balanced labellings and Schubert polynomials, European J. Combin., 8, 373-389, (1997) · Zbl 0871.05059  I.M. Gessel and G.X. Viennot, “Determinants, paths, and plane partitions,” (preprint). · Zbl 0579.05004  R.C. King, “Weight multiplicities for the classical groups,” Springer Lecture Notes in Physics 50(1976). · Zbl 0369.22018  Koike, K.; Terada, I., Young-diagrammatic methods for the representation theory of the classical groups of type B_{n}, C_{n}, D_{n}, J. Algebra, 107, 466-511, (1987) · Zbl 0622.20033  W. KraÜkiewicz and P. Pragacz, “Schubert functors and Schubert polynomials,” 1986 (preprint).  A. Lascoux, “Polynômes de Schubert. Une approche historique,”$$S$$ éries formelles et combinatoire algébrique, P. Leroux and C. Reutenauer (Eds.), Université du Québec à Montréal, LACIM, pp. 283-296, 1992.  I.G. Macdonald, Notes on Schubert polynomials, Laboratoire de combinatoire et d’informatique mathématique (LACIM), Université du Québec à Montréal, Montéral, 1991.  I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Univ. Press, Oxford, 1995.  P.A. MacMahon, Combinatory Analysis, Vols. 1-2, Cambridge University Press, 1915, 1916; reprinted by Chelsea, New York, 1960.  R.A. Proctor, unpublished research announcement, 1984.  Proctor, R. A., Odd symplectic groups, Invent. Math., 92, 307-332, (1988) · Zbl 0621.22009  Proctor, R. A., New symmetric plane partition identities from invariant theory work of De Concini and Procesi, European J. Combin., 11, 289-300, (1990) · Zbl 0726.05008  Reiner, V.; Shimozono, M., Key polynomials and a flagged Littlewood-Richardson rule, J. Combin. Theory, Ser. A, 70, 107-143, (1995) · Zbl 0819.05058  J.-P. Serre, Algebres de Lie Semi-Simples Complexes, W.A. Benjamin, New York, 1966.  Stanley, R. P., Theory and applications of plane partitions, Studies in Appl. Math., 50, 167-188, (1971) · Zbl 0225.05011  Stanley, R. P., On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., 5, 359-372, (1984) · Zbl 0587.20002  Wachs, M. L., Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Combin. Theory, Ser. A, 40, 276-289, (1985) · Zbl 0579.05001  Zhelobenko, D. P., The classical groups. Spectral analysis of their finite dimensional representations, Russ. Math. Surv., 17, 1-94, (1962) · Zbl 0142.26703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.