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Crossings and nestings of matchings and partitions
 Trans. Amer. Math. Soc
"... Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number ..."
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Cited by 61 (15 self)
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Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of knoncrossing partitions is equal to the number of knonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no kcrossing (or with no knesting). 1.
KazhdanLusztig immanants and products of matrix minors
 J. Algebra
, 2006
"... Abstract. We define a family of polynomials of the form ∑ f(σ)x1,σ(1) · · · xn,σ(n) in terms of the KazhdanLusztig basis {C ′ w(1)  w ∈ Sn} for the symmetric group algebra C[Sn]. Using this family, we obtain nonnegativity properties of polynomials of the form ∑ cI,I ′∆I,I′(x)∆I,I′(x). In parti ..."
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Cited by 11 (5 self)
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Abstract. We define a family of polynomials of the form ∑ f(σ)x1,σ(1) · · · xn,σ(n) in terms of the KazhdanLusztig basis {C ′ w(1)  w ∈ Sn} for the symmetric group algebra C[Sn]. Using this family, we obtain nonnegativity properties of polynomials of the form ∑ cI,I ′∆I,I′(x)∆I,I′(x). In particular, we show that the application of certain of these polynomials to JacobiTrudi matrices yields symmetric functions which are nonnegative linear combinations of Schur functions. Résumé. Nous definissons une famille de polynômes ∑ f(σ)x1,σ(1) · · · xn,σ(n) en terme de la base {C ′ w(q)  w ∈ Sn} de KazhdanLusztig pour l’algébra C[Sn]. En utilisant cette famille, nous obtenons quelques propriétées des polynômes totalement non négatifs de la forme ∑ cI,I ′∆I,I′(x)∆I,I′(x). En particulier, nous démontrons que l’application des certains de ces polynômes aux matrices de JacobiTrudi rapporte des fonctions symmétriques qui sont des combinaisons linéaires non négatives des fonctions de Schur. 1.
New branching rules induced by plethysm
 J. Phys A: Math. Gen
, 2006
"... We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi, a symmetric tensor gij = gji an ..."
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Cited by 9 (6 self)
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We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = −fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function sπ ≡ {π} by the basic M series of complete symmetric functions and the L = M −1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras, and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains πgeneralized NewellLittlewood formulae, and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for H 1 3, H21, and H3, showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine, and in some instances non reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly
Geometric approaches to computing Kostka numbers and LittlewoodRichardson coefficients
 PH.D. DEGREE IN MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY (MIT
, 2004
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A unified construction of Coxeter Group representations (III), in preparation. 42
 Discrete Math
"... The goal of this paper is to give a new unified axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras. Building upon fundamental works by Young and KazhdanLusztig, followed by Vershik and Ram, we propose a direct combinatorial construction, avoiding a priori use ..."
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Cited by 3 (1 self)
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The goal of this paper is to give a new unified axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras. Building upon fundamental works by Young and KazhdanLusztig, followed by Vershik and Ram, we propose a direct combinatorial construction, avoiding a priori use of external concepts (such as Young tableaux). This is carried out by a natural assumption on the representation matrices. For simply laced Coxeter groups, this assumption yields explicit simple matrices, generalizing the Young forms. Analysis involves generalized descent classes and convexity (à la Tits) within
DETERMINANT FORMULAS RELATING TO TABLEAUX OF BOUNDED HEIGHT
, 704
"... Abstract. Chen et al. recently established bijections for (d+1)noncrossing/ nonnesting matchings, oscillating tableaux of bounded height d, and oscillating lattice walks in the ddimensional Weyl chamber. Stanley asked what is the total number of such tableaux of length n and of any shape. We find ..."
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Cited by 3 (2 self)
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Abstract. Chen et al. recently established bijections for (d+1)noncrossing/ nonnesting matchings, oscillating tableaux of bounded height d, and oscillating lattice walks in the ddimensional Weyl chamber. Stanley asked what is the total number of such tableaux of length n and of any shape. We find a determinant formula for the exponential generating function. The same idea applies to prove Gessel’s remarkable determinant formula for permutations with bounded length of increasing subsequences. We also give short algebraic derivations for some results of the reflection principle. Mathematics Subject Classification. Primary 05A15, secondary 05A18, 05E10.
The Many Faces Of Modern Combinatorics
, 2003
"... er hand, combinatorial structures are very well suited for experiments using computer algebra systems such as Mathematica and Maple. At this moment, there are a great number of combinatorial packages for the above systems, which enable combinatorialists to gain a lot of insight into their problems, ..."
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er hand, combinatorial structures are very well suited for experiments using computer algebra systems such as Mathematica and Maple. At this moment, there are a great number of combinatorial packages for the above systems, which enable combinatorialists to gain a lot of insight into their problems, by experimenting with their structures up to reasonably high orders of magnitude. Beside computer science, developments within mathematics itself also influenced combinatorics. One of these developments, mentioned in [3], is that after an era where the fashion was to seek generality and abstraction, there is now much appreciation and emphasis for concrete calculations, which are the "hard" problems. The picture of contributions of combinatorics to mathematical calculations can be sketched by referring to three major areas of mathematics. The first two are geometry and topology, with their complicated and hard to manipulate objects. Then comes algebra, which ass