Results 11 to 20 of about 19,463 (248)
The freeness of Shi–Catalan arrangements
European Journal of Combinatorics, 2011 Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H∈A several parallel translations of H by the positive root (and its integer multiples ...
Abe, Takuro +3 more
core +4 more sources
The braid and the Shi arrangements and the Pak–Stanley labelling [PDF]
European Journal of Combinatorics, 2015 In this article we study a construction, due to Pak and Stanley, with which every region RR of the Shi arrangement is (bijectively) labelled with a parking function λ(R). In particular, we construct an algorithm that returns R out of λ(R).
Duarte, Rui +1 more
core +5 more sources
Lattice Point Counts for the Shi Arrangement and other affinographic hyperplane arrangements
Journal of Combinatorial Theory, Series A, 2006 Hyperplanes of the form xj = xi + c are called affinographic. For an affinographic hyperplane arrangement in R n, such as the Shi arrangement, we study the function f(m) that counts integral points in [1, m] n that do not lie in any hyperplane of the ...
Forge, David +3 more
core +3 more sources
Shi arrangements and low elements in affine Coxeter groups
Canadian Journal of Mathematics, 2023 Given an affine Coxeter group $W$, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan-Lusztig cells for $W$. In particular, Shi showed that each region of the
Chapelier-Laget, Nathan +1 more
core +4 more sources
Vertex-weighted Digraphs and Freeness of Arrangements Between Shi and Ish
European Journal of Combinatorics, 2021 We introduce and study a digraph analogue of Stanley's $\psi$-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in literature including the Catalan
Tsujie, Shuhei +2 more
core +2 more sources
A bijection between (bounded) dominant Shi regions and core partitions [PDF]
Discrete Mathematics & Theoretical Computer Science, 2010 International audienceIt is well-known that Catalan numbers $C_n = \frac{1}{ n+1} \binom{2n}{n}$ count the number of dominant regions in the Shi arrangement of type $A$, and that they also count partitions which are both n-cores as well as $(n+1)$-cores.
Vazirani, Monica, Fishel, Susanna
core +5 more sources
Shi arrangements and low elements in Coxeter groups
Proceedings of the London Mathematical SocietyGiven an arbitrary Coxeter system $(W,S)$ and a nonnegative integer $m$, the $m$-Shi arrangement of $(W,S)$ is a subarrangement of the Coxeter hyperplane arrangement of $(W,S)$.
Fishel, Susanna +3 more
core +2 more sources
Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements [PDF]
Journal of Algebraic Combinatorics, 2016 In this article, we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual partition formula. Then, it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl ...
Abe, Takuro, Terao, Hiroaki
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Enumerating regions of Shi arrangements per Weyl Cone
European Journal of Combinatorics, 2023 Given a Shi arrangement $\mathcal{A}_\Phi$, it is well-known that the total number of regions is counted by the parking number of type $\Phi$ and the total number of regions in the dominant cone is given by the Catalan number of type $\Phi$.
Tzanaki, Eleni, Dermenjian, Aram
core +2 more sources
POSETS, PARKING FUNCTIONS AND THE REGIONS OF THE SHI ARRANGEMENT REVISITED
, 2011 The number of regions of the type An−1 Shi arrangement in Rn is counted by the intrinsically beautiful formula (n + 1)n−1. First proved by Shi, this result motivated Pak and Stanley as well as Athanasiadis and Linusson to provide bijective proofs.
Karola Meszaros
core +3 more sources