Hyperplane arrangements between Shi and Ish

We introduce a new family of hyperplane arrangements in dimension n ≥ 3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of this family have the same number of regions — the connected components of the complement of the union of the hyperplanes — which can be bijectively labeled with the Pak-Stanley labelling. In addition, we characterise the Pak-Stanley labels of the regions of this family of hyperplane arrangements.


Introduction
In this paper we introduce a family of arrangements of hyperplanes in general dimension "between Ish and Shi", that is, formed by hyperplanes that are hyperplanes of the Shi arrangement or hyperplanes of the Ish arrangement, all of the same dimension, and characterise the Pak-Stanley labels of the regions of these arrangements of hyperplanes.
Consider, for an integer n ≥ 3, hyperplanes of R n of the following three types. Let, for 1 ≤ i < j ≤ n, Note that C ij | 1 ≤ i < j ≤ n = Cox n is the n-dimensional Coxeter arrangement, A 2 n = Shi n is the n-dimensional Shi arrangement, and A n n = Ish n is the n-dimensional Ish arrangement introduced by Armstrong [2]. We represent both A 2 3 = Shi 3 and A 3 3 = Ish 3 on Figure 1. We study these arrangements with special interest in the Pak-Stanley labelings of the regions of the arrangements. The labels are G-parking functions for special directed multi-graphs G as defined by Mazin [8]. In particular, we show that A k n , for some 2 ≤ k ≤ n, has (n + 1) n−1 regions which are bijectively labeled.
The notion of G-parking function was introduced by Postnikov and Shapiro in the construction of two algebras related to a general undirected graph G [9]. Later, Hopkins and Perkinson [6] showed that the labels of the Pak-Stanley labeling of the regions of a given hyperplane arrangement defined by G are exactly the G-parking functions, a fact that had been conjectured by Duval, Klivans and Martin [5]. Recently, Mazin [8] generalized this result to a very general class of hyperplane arrangements, with a similar concept based on a general directed multi-graph G. Whereas Hopkins and Perkinson's hyperplane arrangements include for example the (original) multidimensional Shi arrangement, Mazin's hyperplane arrangements include the multidimensional k-Shi arrangement, the multidimensional Ish arrangement and in fact all the arrangements we consider here.

The characteristic polynomial
In the following, we evaluate the characteristic polynomial χ(A k n , q).
In terms of the arrangements A k n , this means, by a celebrated result of Zaslavsky [12], that the number of regions, r(A k n ), and the number of relatively bounded regions, b(A k n ), do not depend on 2 ≤ k ≤ n, being These results were known (and proven similarly [1]) for both the Shi and the Ish arrangements [3,7], although not for the remaining arrangements A k n , with 2 < k < n. Similarly to what Pak and Stanley did for the regions of the Shi arrangement (cf. [11]), we may represent a region R of A = A k n as follows. Suppose that x = (x 1 , . . . , x n ) ∈ R and x w 1 > · · · > x wn for a given w = (w 1 , . . . , w n ) ∈ S n . Let H be the set of triples (i, j, a ij ) such that i, j, a ij ∈ N, 1 ≤ i < j ≤ n, x i > x j and a ij − 1 < x i − x j < a ij . Then, of course, We represent R by w, decorated with one labelled arc for each hyperplane of H, as follows. Given (i, j, a ij ) ∈ H, the arc connects i with j and is labelled a ij , with the following exceptions: if i ≤ j < p ≤ m, (i, m, a im ), (j, p, a jp ) ∈ H and a jp = a im , then we omit the arc connecting j with p. In the Figure 2 the regions of Shi 3 and Ish 3 are thus represented.
The Pak-Stanley labelling of these regions may be defined as follows. As usual, let e i = (0, . . . , 0, 1, 0, . . . , 0) be the i.th element of the standard basis of R n .  , 1), and, given two regions R 1 and R 2 separated by a unique hyperplane H of A k n such that R 0 and R 1 are on the same side of H, label the regions R 1 and R 2 so that In the Figure 1 the Pak-Stanley labellings of the regions of Shi 3 and Ish 3 are shown.

Partial Parking Functions
The labels of the regions of the Shi arrangement A 2 n = Shi n form the set of n-dimensional parking functions, defined below, as proven by Pak and Stanley in their seminal work [10]. permutation π ∈ S n such that By labelling under the same rules the regions of the n-dimensional Ish arrangement A n n = Ish n , we obtain the Ish-parking functions [4] which can be characterised as follows. Proposition 4.3 a = (a 1 , . . . , a n ) ∈ [n] n is an Ish-parking function if and only if 1 ∈ Z(a).
Finally, we show that the sets of labels corresponding to the arrangements A k n (2 ≤ k ≤ n) that interpolate between the Shi and the Ish arrangements are the k-partial parking functions which are characterised as follows.