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104
NonCrossing Partitions For Classical Reflection Groups
 Discrete Math
, 1996
"... We introduce analogues of the lattice of noncrossing set partitions for the classical reflection groups of type B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as wellbehaved as the original noncrossing set partitions, and the type D analogues ..."
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Cited by 115 (5 self)
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We introduce analogues of the lattice of noncrossing set partitions for the classical reflection groups of type B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as wellbehaved as the original noncrossing set partitions, and the type D analogues almost as wellbehaved. In both cases, they are ELlabellable ranked lattices with symmetric chain decompositions (selfdual for type B), whose rankgenerating functions, zeta polynomials, rankselected chain numbers have simple closed forms.
Permutations with restricted patterns and Dyck paths
 Adv. Appl. Math
"... Abstract. We exhibit a bijection between 132avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132avoiding permutations with a given number of occurrences of the pattern 12... k follow directly from old resul ..."
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Cited by 99 (3 self)
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Abstract. We exhibit a bijection between 132avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132avoiding permutations with a given number of occurrences of the pattern 12... k follow directly from old results on the enumeration of Motzkin paths, among which is a continued fraction result due to Flajolet. As a bonus, we use these observations to derive further results and a precise asymptotic estimate for the number of 132avoiding permutations of {1, 2,..., n} with exactly r occurrences of the pattern 12... k. Second, we exhibit a bijection between 123avoiding permutations and Dyck paths. When combined with a result of Roblet and Viennot, this bijection allows us to express the generating function for 123avoiding permutations with a given number of occurrences of the pattern (k − 1)(k − 2)...1k in form of a continued fraction and to derive further results for these permutations.
Generating Trees and Forbidden Subsequences
, 1996
"... this paper all produce enumerative results which, to the best of our knowledge, were not previously known: ..."
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Cited by 72 (2 self)
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this paper all produce enumerative results which, to the best of our knowledge, were not previously known:
Gröbner geometry of Schubert polynomials
 Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
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Cited by 68 (14 self)
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Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are therefore positive sums of monomials, each monomial representing the torusequivariant cohomology class of a component (a schemetheoretically reduced coordinate subspace) in the limit of the resulting Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant Ktheory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in Ktheory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces is provided by the proof of the Gröbner basis property, bypassing what has come to be known as Kohnert’s conjecture [Mac91]. The coordinate subspaces, which are
The Enumeration of Fully Commutative Elements of Coxeter Groups
, 1996
"... this paper, we consider the problem of enumerating the fully commutative elements of these groups. The main result (Theorem 2.6) is that for six of the seven infinite families (we omit the trivial dihedral family I 2 (m)), the generating function for the number of fully commutative elements can be e ..."
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Cited by 62 (4 self)
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this paper, we consider the problem of enumerating the fully commutative elements of these groups. The main result (Theorem 2.6) is that for six of the seven infinite families (we omit the trivial dihedral family I 2 (m)), the generating function for the number of fully commutative elements can be expressed in terms of three simpler generating functions for certain formal languages over an alphabet with at most four letters. The languages in question vary from family to family, but have a uniform description. The resulting generating function one obtains for each family is algebraic, although in some cases quite complicated. (See (3.7) and (3.11).) In a general Coxeter group, the fully commutative elements index a basis for a natural quotient of the corresponding IwahoriHecke algebra [G]. (See also [F1] for the simplylaced case.) For An , this quotient is the TemperleyLieb algebra. Recently, Fan [F2] has shown that for types A, B, D, E and (in a sketched proof) F , this quotient is generically semisimple, and gives recurrences for the dimensions of the irreducible representations. (For type H, the question of semisimplicity remains open.) This provides another possible approach to computing the number of fully commutative elements in these cases; namely, as the sum of the squares of the dimensions of these representations. Interestingly, Fan also shows that the sum of these dimensions is the number of fully commutative involutions
Schubert Polynomials for the Classical Groups
 J. Amer. Math. Soc
, 1994
"... Introduction The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special ..."
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Cited by 55 (4 self)
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Introduction The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special case, the Schubert calculus for Grassmannians, where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions. A typical problem is to find the number of lines meeting four given lines in general position in 3space (answer below). For each of the four given lines, the set of lines meeting it is a Schubert variety in the Grassmannian and we want the number of intersection points of these four subvarieties. In the modern solution of this problem, the Schubert varieties induce canonical elements of the cohomology ring of the Grassmannian, called Schubert classes. The product of these Schubert classes is the class of a point times the number of inter
A LittlewoodRichardson rule for the Ktheory of Grassmannians
 Acta Math
"... Abstract. We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate Ktheory of Grassmannians to a bialgebra of stable Grothendieck polynomials, which is a Kth ..."
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Cited by 41 (0 self)
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Abstract. We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate Ktheory of Grassmannians to a bialgebra of stable Grothendieck polynomials, which is a Ktheory parallel of the ring of symmetric functions. 1.
Pieri's Formula For Flag Manifolds And Schubert Polynomials
 Ann. Inst. Fourier (Grenoble
, 1996
"... . We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the c ..."
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Cited by 41 (18 self)
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. We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri's formula for symmetric polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, some of which we express in terms of paths in the Bruhat order on the symmetric group, which in turn yields an enumerative result about the Bruhat order. R'esum'e Nous 'etablissons la formule pour la multiplication par la classe d'une vari'et'e de Schubert sp'eciale dans l'anneau de cohomologie de la vari'et'e de drapeaux. Cette formule d'ecrit aussi la multiplication d'un polynome de Sch...
RCGraphs and Schubert Polynomials
, 1993
"... CONTENTS 1. Introduction 2. Background on Schubert Polynomials 3. Constructing Schubert Polynomials From RCGraphs 4. Double Schubert Polynomials 5. Monk's Rule 6. Conjectures Acknowledgements References Bergeron was supported by the National Science Foundation. Billey was supported by the National ..."
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Cited by 39 (5 self)
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CONTENTS 1. Introduction 2. Background on Schubert Polynomials 3. Constructing Schubert Polynomials From RCGraphs 4. Double Schubert Polynomials 5. Monk's Rule 6. Conjectures Acknowledgements References Bergeron was supported by the National Science Foundation. Billey was supported by the National Physical Science Consortium, IBM and UCSD. Using a formula of Billey, Jockusch and Stanley, Fomin and Kirillov have introduced a new set of diagrams that encode the Schubert polynomials. We call these objects rcgraphs. We define and prove two variants of an algorithm for constructing the set of all rcgraphs for a given permutation. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we give a new proof of Monk's rule using an insertion algorithm on rcgraphs. We conjecture two analogs of Pieri's rule for multiplying Schubert polynomials. We also extend the algorithm to generate the double
Noncommutative Schur Functions and their Applications
"... We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many wellknown associative algebras. As an application of our theory, we prove Schurpositivity and obtain generalized LittlewoodRichardson and MurnaghanNakayama rules for a larg ..."
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Cited by 38 (1 self)
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We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many wellknown associative algebras. As an application of our theory, we prove Schurpositivity and obtain generalized LittlewoodRichardson and MurnaghanNakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials. 1. Introduction and Main Results In this paper we develop a theory of Schur functions in noncommuting variables, assuming certain commutation relations that are satisfied in many wellknown examples such as the plactic, nilplactic, and nilCoxeter algebras and the degenerate Hecke algebra Hn (0). We show that most of the classical theory of symmetric functions can be reproduced in this noncommutative setting. There are many combinatorial representations of these commutation relations, and to each of these one can associate a family of (ordinary) symmetric functions; examples of such families include skew ...