Between Shi and Ish

Duarte, Rui; Oliveira, António Guedes de

doi:10.1016/j.disc.2017.09.006

Cited by 7 publications

(8 citation statements)

References 12 publications

(84 reference statements)

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“…(2) In the right-hand side of Figure 1 the Pak-Stanley labelling of the regions of Ish 3 is shown. In dimension n, these labels form the set of n-dimensional Ish-parking functions, characterized in a previous article [6]. The labels of the regions of Shi n form the set of n-dimensional parking functions, defined below, as proven by Pak and Stanley in their seminal work [12].…”

Section: Note Thatmentioning

confidence: 97%

“…Figure 3). Recall that this algorithm, given a ∈ [n] n and a multiple digraph G, determines whether a is a G-parking function by constructing in the positive case an oriented spanning subtree T of G that is in bijection with a [10,6]. The Tree to Parking Function Algorithm (cf.…”

Section: The Dfs-burning Algorithmmentioning

confidence: 99%

“…In this paper, we characterise the Pak-Stanley labels of the regions of the recently introduced family of the arrangements of hyperplanes "between Shi and Ish" (cf. [6]).…”

Section: Introductionmentioning

confidence: 99%

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Partial parking functions

Duarte

Oliveira

2019

Discrete Mathematics

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Section: Note Thatmentioning

confidence: 97%

Section: The Dfs-burning Algorithmmentioning

confidence: 99%

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Partial parking functions

Duarte

Oliveira

2019

Discrete Mathematics

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“…In the classical case of extended Shi arrangements, one can show the bijectivity of the Pak-Stanley labeling by using the above results and then comparing the cardinalities of the two sets. The bijectivity results can be extended to other families of arrangements (see [2]). However, in general the generalized Pak-Stanley labelings often fail to be injective.…”

Section: Introductionmentioning

confidence: 99%

Pak-Stanley labeling for Central Graphical Arrangements

Mazin¹,

Miller²

2018

Preprint

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The original Pak-Stanley labeling was defined by Pak and Stanley in [5] as a bijective map from the set of regions of an extended Shi arrangement to the set of parking functions. This map was later generalized to other arrangements associated with graphs and directed multigraphs (see [1,3,4]). In these more general cases the map is no longer bijective. However, it was shown in [3] and [4] that it is surjective to the set of the G-parking functions, where G is the multigraph associated with the arrangement.This leads to a natural question: when is the generalized Pak-Stanley map bijective? In this paper we answer this question in the special case of centered hyperplane arrangements, i.e. the case when all the hyperplanes of the arrangement pass through a common point.

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“…In this paper we will focus on a larger family containing Shi(ℓ) and Ish(ℓ), the arrangements "between Shi and Ish" introduced recently by Duarte and Guedes de Oliveira in their study of Pak-Stanley labeling of the regions of real arrangements [23,24].…”

Section: Introductionmentioning

confidence: 99%

Vertex-weighted Digraphs and Freeness of Arrangements Between Shi and Ish

Abe¹,

Tran²,

Tsujie³

2021

Preprint

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We introduce and study a digraph analogue of Stanley's ψ-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 arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish recently introduced by Duarte and Guedes de Oliveira. The arrangements between Shi and Ish all are proved to have the same characteristic polynomial with all integer roots, thus raising the natural question of their freeness. We define two operations on digraphs, which we shall call king and coking elimination operations and prove that subject to certain conditions on the weight ψ, the operations preserve the characteristic polynomials and freeness of the associated arrangements. As an application, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone.

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Between Shi and Ish

Cited by 7 publications

References 12 publications

Partial parking functions

Partial parking functions

Pak-Stanley labeling for Central Graphical Arrangements

Vertex-weighted Digraphs and Freeness of Arrangements Between Shi and Ish

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