Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes

By Christos A. Athanasiadis. 2004

93 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/765bcfec107353726aca5e05062ebaf06e6cd63b

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Citations in the corpus

Characterization of a free arrangement and conjecture of Edelman and Reiner (2004, methodology): “And their combinatorics was investigated by many people [Sta], [Ath1], [ER], [He], [PS], [ Ath4 ].”
On a refinement of the generalized Catalan numbers for Weyl groups (2004, background): “In the special case m = 1, the expression (1.1) is refered to as the Catalan number associated to [ 2 , 17], since it reduces to the familiar n th Catalan number for the root system An 1. It has a variety of interesting combinatorial interpretations apart from those given in Theorem 1.1, for instanc”
A uniform bijection between nonnesting and noncrossing partitions (2011, background): “stem of polynomial invariants for W(called the degrees of W), and where his the Coxeter number. This formula was ﬁrst conjectured by Postnikov [19, Remark 2] and at least two uniform proofs are known [2, 12]. These enumerations were established in somewhat diﬀerent contexts; the link to the combinat”
Root systems and generalized associahedra (2005, background): “We refer the reader to the introductory sections of [1, 2, 3, 12, 39] for the history of research in this area, for further details and references, and for numerous generalizations and connections.”
B-Stable Ideals in the Nilradical of a Borel Subalgebra (2003, unknown): “Finally if αl is short or αj is long and αj 6= −αl, then 〈αl, α∨j 〉 = −1; hence there exist m 6= l with 〈αm, α ∨ j 〉 < 0, implying that also αm + αj ∈ Φ ∪ {0}.”
Freeness of hyperplane arrangements and related topics (2012, unknown): “on theorem (Theorem 1.17), we have the following. Corollary 3.2. (i) χ(A[−k,k] Φ ,t) = Qℓ i=1(t −ei −kh). (ii) χ(A[1−k,k] Φ ,t) = (t− kh)ℓ. In the above corollary, (i) has been proved by Athanasiadis [9] by a purely combinatorial method. Edelman-Reiner [12] and Headley [17] proved (ii) for some spec”