Related papers: Bijectivity of a generalized Pak-Stanley labeling
The original Pak-Stanley labeling was defined by Pak and Stanley 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…
This article has the sole purpose of presenting a simple, self-contained and direct proof of the fact that the Pak-Stanley labeling is a bijection. The construction behind the proof is subsumed in a forthcoming paper [R. Duarte and A.…
Back in the nineties Pak and Stanley introduced a labeling of the regions of a k-Shi arrangement by k-parking functions and proved its bijectivity. Duval, Klivans, and Martin considered a modification of this construction associated with a…
It is known that the Pak-Stanley labeling of the Shi hyperplane arrangement provides a bijection between the regions of the arrangement and parking functions. For any graph G, we define the G-semiorder arrangement and show that the…
We characterise the Pak-Stanley labels of the regions of a family of hyperplane arrangements that interpolate between the Shi arrangement and the Ish arrangement.
We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson.…
Given a simple graph $G$, one can define a hyperplane arrangement called the $G$-Shi arrangement. The Pak-Stanley algorithm labels the regions of this arrangement with $G_\bullet$-parking functions. When $G$ is a complete graph, we recover…
Associated with the $r$-Shi arrangement and $r$-Catalan arrangement in $\Bbb{R}^n$, we introduce a cubic matrix for each region to establish two bijections in a uniform way. Firstly, the positions of minimal positive entries in column…
We characterize in simple terms the Pak-Stanley labels $\lambda(R)$ of the regions $R$ of the $m$-Catalan arrangement. We also propose a simple algorithm that returns $R$ from $\lambda(R)$. Finally, we characterize in close terms the labels…
The Shi arrangement ${\mathcal S}_n$ is the arrangement of affine hyperplanes in ${\mathbb R}^n$ of the form $x_i - x_j = 0$ or $1$, for $1 \leq i < j \leq n$. It dissects ${\mathbb R}^n$ into $(n+1)^{n-1}$ regions, as was first proved by…
There are several combinatorial objects that are known to be in bijection to the spanning trees of a graph G. These objects include G-parking functions, critical configurations of G, and descending traversals of G. In this paper, we extend…
We introduce a new family of hyperplane arrangements in dimension $n\geq3$ that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions - the connected…
The \emph{Shi arrangement} is the set of all hyperplanes in $\mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 \le j < k \le n$. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this…
We establish a general bijective framework for encoding faces of some classical hyperplane arrangements. Precisely, we consider hyperplane arrangements in $\mathbb{R}^n$ whose hyperplanes are all of the form $\{x_i-x_j=s\}$ for some…
Motivated by the relation holding for the m-generalized Catalan numbers of type A and C, the connection between dominant regions of the m-Shi arrangement of type A and C is investigated. In the same line of thought, a bijection between mn+1…
We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some…
Combining Carver's variant of the Farkas' lemma with the Flow Decomposition Theorem we show that the regions of any deformation of a graphical arrangement may be bijectively labeled with a set of weighted digraphs containing directed cycles…
It 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. These…
We introduce type C parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type C parking functions to regions of the Shi arrangement of type C, encoded as diagonally…
The number of regions of the type A_{n-1} Shi arrangement in R^n 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…