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\(\sum\limits_{(a_1 ,a_2 , \ldots ) \in Red(w_0 )} {(x + a_1 )(x + a_2 )} \cdots = \left( {_2^n } \right)!\prod\limits_{1 \leqslant i < j \leqslant n} {\frac{{2x + i + j - 1}}{{i + j - 1}}} ,\)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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.C. Billey, W. Jockusch, and R.P. Stanley, “Some combinatorial properties of Schubert polynomials,” J. Alg. Combin. 2(1993), 345–374.
P. Edelman and C. Greene, “Balanced tableaux,” Advances in Math. 63(1987), 42–99.
S. Fomin and R.P. Stanley, “Schubert polynomials and the nilCoxeter algebra,” Advances in Math. 103(1994), 196–207.
S. Fomin and A.N. Kirillov, “The Yang-Baxter equation, symmetric functions, and Schubert polynomials,” Discrete Math. 153(1996), 123–143.
S. Fomin, C. Greene, V. Reiner, and M. Shimozono, “Balanced labellings and Schubert polynomials,” European J. Combin. 8(1997), 373–389.
I.M. Gessel and G.X. Viennot, “Determinants, paths, and plane partitions,” (preprint).
R.C. King, “Weight multiplicities for the classical groups,” Springer Lecture Notes in Physics 50(1976).
K. Koike and I. Terada, “Young-diagrammatic methods for the representation theory of the classical groups of type B n , C n , D n ,” J. Algebra 107(1987), 466–511.
W. KraÜkiewicz and P. Pragacz, “Schubert functors and Schubert polynomials,” 1986 (preprint).
A. Lascoux, “Polynômes de Schubert. Une approche historique,” Séries 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.
R.A. Proctor, “Odd symplectic groups,” Invent. Math. 92(1988), 307–332.
R.A. Proctor, “New symmetric plane partition identities from invariant theory work of De Concini and Procesi,” European J. Combin. 11(1990), 289–300.
V. Reiner and M. Shimozono, “Key polynomials and a flagged Littlewood-Richardson rule,” J. Combin. Theory, Ser. A 70(1995), 107–143.
J.-P. Serre, Algebres de Lie Semi-Simples Complexes, W.A. Benjamin, New York, 1966.
R.P. Stanley, “Theory and applications of plane partitions,” Studies in Appl. Math. 50(1971), 167–188, 259–279.
R.P. Stanley, “On the number of reduced decompositions of elements of Coxeter groups,” European J. Combin. 5(1984), 359–372.
M.L. Wachs, “Flagged Schur functions, Schubert polynomials, and symmetrizing operators,” J. Combin. Theory, Ser. A 40(1985), 276–289.
D.P. Zhelobenko, “The classical groups. Spectral analysis of their finite dimensional representations,” Russ. Math. Surv. 17(1962), 1–94.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fomin, S., Kirillov, A.N. Reduced Words and Plane Partitions. Journal of Algebraic Combinatorics 6, 311–319 (1997). https://doi.org/10.1023/A:1008694825493
Issue Date:
DOI: https://doi.org/10.1023/A:1008694825493