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55
Enumerative Geometry For The Real Grassmannian Of Lines In Projective Space
 Duke Math. J
, 1996
"... Given Schubert conditions on lines in projective space which generically determine a finite number of lines, we show there exist general real conditions determining the expected number of real lines. This extends the classical Schubert calculus of enumerative geometry for the Grassmann variety of ..."
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Cited by 30 (16 self)
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Given Schubert conditions on lines in projective space which generically determine a finite number of lines, we show there exist general real conditions determining the expected number of real lines. This extends the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Our main tool is an explicit geometric description of rational equivalences, which also constitutes a novel determination of the Chow rings of these Grassmann varieties of lines.
A Direct Bijective Proof of the HookLength Formula.
, 1997
"... This paper presents a new proof of the hooklength formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity o ..."
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Cited by 16 (0 self)
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This paper presents a new proof of the hooklength formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. We finish with some examples. 1
Combinatorics, Symmetric Functions, and Hilbert Schemes
 CDM 2002: Current Developments in Mathematics, Intl
, 2003
"... We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n + 1) " conjectures relating Macdonald polynomials to the c ..."
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Cited by 15 (0 self)
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We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n + 1) " conjectures relating Macdonald polynomials to the characters of doublygraded Sn modules. To make the treatment selfcontained, we include background material from combinatorics, symmetric function theory, representation theory and geometry. At the end we discuss future directions, new conjectures and related work of Ginzburg, Kumar and Thomsen, Gordon, and Haglund and Loehr.
Combinatorial representation theory
 in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–97), MSRI Publ. 38
, 1999
"... Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when ..."
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Cited by 13 (0 self)
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Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when known. Background material and more specialized results are given in a series of appendices. We give a personal view of the field while remaining aware that there is much important and beautiful work that we have been unable to mention.
Another involution principlefree bijective proof of Stanley’s hookcontent formula
 J. Combin. Theory Ser. A
, 1999
"... Abstract. Another bijective proof of Stanley’s hookcontent formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution principle of Garsia and Milne. It is the result of a merge of the modified jeu de taquin idea from the author’s pre ..."
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Cited by 12 (3 self)
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Abstract. Another bijective proof of Stanley’s hookcontent formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution principle of Garsia and Milne. It is the result of a merge of the modified jeu de taquin idea from the author’s previous bijective proof (“An involution principlefree bijective proof of Stanley’s hookcontent formula”, Discrete Math. Theoret. Computer Science, to appear) and the NovelliPakStoyanovskii bijection (Discrete Math. Theoret. Computer Science 1 (1997), 53–67) for the hook formula for standard Young tableaux of a given shape. This new algorithm can also be used as an algorithm for the random generation of tableaux of a given shape with bounded entries. An appropriate deformation of this algorithm gives an algorithm for the random generation of plane partitions inside a given box. 1. Introduction. There
Hook Length Formula and Geometric Combinatorics
 Lothar. Comb
, 2000
"... We present here a transparent proof of the hook length formula. The formula is reduced to an equality between the number of integer point in certain polytopes. The latter is established by an explicit continuous volumepreserving piecewise linear map. ..."
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Cited by 11 (7 self)
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We present here a transparent proof of the hook length formula. The formula is reduced to an equality between the number of integer point in certain polytopes. The latter is established by an explicit continuous volumepreserving piecewise linear map.
Discovering hook length formulas by expansion technique
"... ABSTRACT. — We introduce a hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and ..."
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Cited by 11 (4 self)
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ABSTRACT. — We introduce a hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook length expansion. The paper can be seen as a collection of hook length formulas for partitons and plane trees. All examples are illustrated by HookExp and, for many easy cases, expained by wellknown combinatorial arguments. Summary §1. Introduction. Selected hook formulas. Conjecture §2. Classical hook length formulas for partitions. §3. Hook length expansion algorithm and HookExp. §4. The exponent principle. §5. Hook length formulas for partitions.
Transcendence of Formal Power Series With Rational Coefficients
, 1999
"... We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal po ..."
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Cited by 9 (2 self)
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We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal power series, binomial series, automatic sequences, pLucas sequences, ChomskySchutzenberger theorem. 1 Introduction Formal power series with integer coefficients often occur as generating series. Suppose that a set E contains exactly a n elements of "size" n for each integer n: the generating series of this set is the formal power series P n0 a n X n (this series belongs to Z[[X]] ae Q[[X]]). Properties of this formal power series reflect properties of its coefficients, and hence properties of the set E. Roughly speaking, algebraicity of the series over Q(X) means that the set has a strong structure. For example, the ChomskySchutzenberger theorem [16] asserts that the generating seri...
Bijective proofs of the hook formulas for the number of standard Young tableaux, ordinary and shifted
, 1995
"... . Bijective proofs of the hook formulas for the number of ordinary standard Young tableaux and for the number of shifted standard Young tableaux are given. They are formulated in a uniform manner, and in fact prove qanalogues of the ordinary and shifted hook formulas. The proofs proceed by combinin ..."
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Cited by 9 (0 self)
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. Bijective proofs of the hook formulas for the number of ordinary standard Young tableaux and for the number of shifted standard Young tableaux are given. They are formulated in a uniform manner, and in fact prove qanalogues of the ordinary and shifted hook formulas. The proofs proceed by combining the ordinary, respectively shifted, HillmanGrassl algorithm and Stanley's (P; !)partition theorem with the involution principle of Garsia and Milne. 1. Introduction. A few years ago there had been a lot of interest in finding a bijective proof of Frame, Robinson and Thrall's [1] hook formula for the number of standard Young tableaux of a given shape. This resulted in the discovery of three different such proofs [2, 10, 14], none of them is considered to be really satisfactory. Closest to being satisfactory is probably the proof by Franzblau and Zeilberger [2]. However, while the description of their algorithm is fairly simple, it is rather difficult to show that it really works. Also, i...
Cyclic sieving, promotion, and representation theory
, 2008
"... Abstract. We prove a collection of conjectures of D. White [37], as well as some related conjectures of AbuzzahabKorsonLiMeyer [1] and of Reiner and White [21], [37], regarding the cyclic sieving phenomenon of Reiner, Stanton, and White [22] as it applies to jeudetaquin promotion on rectangular ..."
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Cited by 9 (0 self)
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Abstract. We prove a collection of conjectures of D. White [37], as well as some related conjectures of AbuzzahabKorsonLiMeyer [1] and of Reiner and White [21], [37], regarding the cyclic sieving phenomenon of Reiner, Stanton, and White [22] as it applies to jeudetaquin promotion on rectangular tableaux. To do this, we use KazhdanLusztig theory and a characterization of the dual canonical basis of C[x11,...,xnn] due to Skandera [27]. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups and noncrossing partitions.