The Shi arrangement and the Ish arrangement

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious"combinatorial symmetry"between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$"ceilings"and $d$"degrees of freedom", etc. Moreover, all of these results hold in the greater generality of"deleted"Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.


Introduction
A hyperplane arrangement is a finite collection of affine hyperplanes in Euclidean space. Some of the nicest arrangements come from the reflecting hyperplanes of Coxeter groups. In particular, the Coxeter arrangement of type A (also known as the braid arrangement) is the arrangement in R n defined by Cox(n) := {x i − x j = 0 : 1 ≤ i < j ≤ n}.
Here {x 1 , . . . , x n } are the standard coordinate functions on R n . Postnikov and Stanley [9] introduced the idea of a deformation of the Coxeter arrangement -this is an affine arrangement each of whose hyperplanes is parallel to some hyperplane of the Coxeter arrangement. In the present paper we will study two specific deformations of the Coxeter arrangement and we will observe a deep similarity between them. The first is the Shi arrangement which was one of Postnikov and  This arrangement was defined by Jian-Yi Shi [11,Chapter 7] in the study of the Kazhdan-Lusztig cellular structure of the affine Weyl group of type A. The second is the Ish arrangement, recently defined by the first author [1]: He used the Shi and Ish arrangements to give a new description of the q, t-Catalan numbers of Garsia and Haiman in terms of the affine Weyl group of type A. Figure  1.1 displays the arrangements Shi(3) and Ish (3). (Note that the normals to the hyperplanes of either Shi(n) or Ish(n) span the hyperplane x 1 + x 2 + · · · + x n = 0. Hence we will always draw their restrictions to this space.) The heart of this paper is the following correspondence between Shi and Ish hyperplanes. The correspondence is natural to state but we find it geometrically mysterious. We will call this a "combinatorial symmetry": This symmetry allows us to define deleted versions of the Shi and Ish arrangements. Let The arrangement Shi(G) was first considered by Athanasiadis [3]. Note that Shi(G) (resp. Ish(G)) interpolates between the Coxeter arrangement and the Shi (resp. Ish)   [3] 2 . In order to state our main results right away we need a few definitions.
Let A be either Shi(G) or Ish(G). The connected components of R n − ∪ H∈A H are called regions. We say that a region is dominant if it lies within the dominant cone, defined by the coordinate inequalities The topological closureR of a region R is decomposed by the arrangement A into faces of various dimensions. We say that the hyperplane H is a wall of R if it is the affine span of a codimension-1 face of R. The wall H is called a ceiling if H does not contain the origin and if the region R and the origin lie in the same half-space of H. Since every region R is convex, it determines a recession cone (which is closed under non-negative linear combinations): Rec(R) := {v ∈ R n : v + R ⊆ R}.
Note that the region R is bounded if and only if Rec(R) = 0. We call the dimension of Rec(R) the number of degrees of freedom of R. This poset has the structure of a geometric semilattice (see [14]) with a unique minimum element R n (corresponding to the empty intersection). The characteristic polynomial (or chromatic polynomial) χ A (p) ∈ Z[p] of the arrangement A is defined by where µ : L(A) × L(A) → Z is the Möbius function of the poset L(A) (see [12]).
be a simple loopless graph on n vertices; let c and d be nonnegative integers. The deleted Shi and Ish arrangements Shi(G) and Ish(G) share the following properties in common: (1) the characteristic polynomial;  We find it surprising that the symmetry x i −x j = 1 ↔ x 1 −x j = i preserves so much information. However, there are important properties that it does not preserve. For example, one may observe from Figures 1.1 and 1.2 that the intersection poset is not preserved. One can also show that the Tutte polynomials of Shi(3) and Ish(3) differ, and that the Orlik-Solomon algebras of Shi(G) and Ish(G) are not graded-isomorphic for G = {12, 23} (even though the equality of characteristic polynomials implies that these algebras have the same Hilbert series). Is there a unifying concept that could simplify the statement of the Main Theorem?
The paper is structured as follows.
In Section 2 we establish some language for set partitions. We define G-partitions -which for the complete graph are just partitions of [n] = {1, 2, . . . , n} -and discuss various kinds: connected, nonnesting. We define the (a, b) endpoint notation for partitions which seems to be the correct language for comparing Shi and Ish arrangements.
In Section 3 we show that Shi(G) and Ish(G) have the same characteristic polynomial, which has a formula involving G-Stirling numbers. This proves part (1) of the Main Theorem. Our tool is the finite field method Crapo and Rota [5]. By a standard result of Zaslavsky this implies that Shi(G) and Ish(G) share the same numbers of total regions and relatively bounded regions (regions with one degree of freedom).
In Section 4 we modify a labeling of the regions of Shi(G) due to Athanasiadis and Linusson [2] and we call the result Shi ceiling diagrams. Similarly, we define Ish ceiling diagrams for the regions of Ish(G). We give a bijective proof of part (2) of the Main Theorem by observing that dominant regions of Shi(G) and Ish(G) correspond to order ideals in isomorphic posets.
In Section 5 we define the ceiling partition for a region of Shi(G) or Ish(G). This is a (possibly nesting) G-partition that encodes the ceilings of the region. Given a G-partition π with k blocks and an integer 1 ≤ d ≤ k, we show that the number of regions of either Shi(G) or Ish(G) with ceiling partition π and d degrees of freedom is equal to which proves part (3) of the Main Theorem. This formula is remarkable, and it is new even for the Shi arrangement. The proof of the formula for Ish regions is direct, whereas the proof for Shi regions uses a new formula due to the second author (see [10] or We end with an observation: The Ish arrangement is something of a "toy model" for the Shi arrangement (and other Catalan objects). That is, for any property P that Shi(G) and Ish(G) share, the proof that Ish(G) satisfies P is easier than the proof that Shi(G) satisfies P .

Set Partitions
All of the formulas in this paper are phrased in terms of set partitions. In this section we will give some background on these and establish notation. In particular, for each graph G ⊆   [8] with type (1, 0, 1, 1, 0, 0, 0, 0) 2.1. The endpoint notation. We say that π = {B 1 , B 2 , . . . , B k } is a partition of [n] into k blocks if the following disjoint union holds: The type of the partition π is the sequence (r 1 , r 2 , . . . , r n ) where r i is the number of blocks of π with size i. We draw the arc diagram of π as follows: Place the numbers 1, 2, . . . , n on a line and draw an arc between each pair i < j such that • i and j are in the same block of π; and • there is no i < ℓ < j such that i, ℓ, j are in the same block of π. In this paper we will use a special notation for partitions, based on the arc diagram. First note that a partition π has n − k blocks if and only if its diagram has k arcs. This is because each new arc reduces the number of blocks by one. Now suppose that the arcs of π are a 1 b 1 , a 2 b 2 , . . . , a k b k , with the left endpoints in increasing order: a 1 < a 2 < · · · < a k . We will associate π with its pair (a, b) of endpoint vectors: We call (a, b) the endpoint notation for π. For example, the endpoint notation for the partition in Figure  We will see that the endpoint notation is the best language for comparing Shi and Ish arrangements.

Nonnesting partitions.
A partition π of [n] is called nonnesting if it does not contain arcs ij and kℓ such that i < k < ℓ < j -that is, no arc of π "nests" inside another. The partition in Figure 2.1 is not nonnesting (it is nesting) because the arc 56 nests inside the arc 37. The number of nonnesting partitions of [n] is famously given by the Catalan number 1 n+1 2n n . The property of nonnesting agrees well with the endpoint notation for partitions. That is, a partition (a, b) is nonnesting if and only if its right endpoint vector b is increasing. In fact, the number of pairs of nesting arcs in (a, b) is equal to the number of pairs b i > b j such that i < j.

2.4.
Connectivity. Finally, we mention an auxiliary (nontrivial) result which we need for the proof of the Main Theorem. For i ≤ j we say that a partition π of the set {i, i + 1, . . . , j} is connected if there does not exist i ≤ k < j such that π refines the partition {{i, i + 1, . . . , k}, {k + 1, . . . , j − 1, j}}.
Equivalently, π is connected if its arc diagram has no holes when seen from space. The partition in Figure 2.1 is connected. Moreover, a partition π of [n] has d connected components if there exist numbers 1 < i 1 , < · · · < i d−1 < n such that π refines the partition and if its restriction to each block of this partition is connected. Equivalently, the arc diagram of a partition with d connected components has d − 1 holes when seen from space. For example, the partition {{1, 2}, {3, 5, 7}, {4, 6}, {8}} has 3 connected components.
The second author has recently established an enumerative formula for nonnesting partitions (and other Catalan objects) that takes account of the type of the partition and its number of connected components. The prototype for this formula is the following theorem of Kreweras [7,Theorem 4]. Kreweras stated his formula in terms of noncrossing partitions; however, type-preserving bijections between noncrossing and nonnesting partitions have been observed by several authors. Lemma 2.1. Let n > 0 and suppose that the sequence (r 1 , . . . , r n ) of nonnegative integers satisfies i ir i = n and i r i = k. The number of nonnesting partitions of We will need the following formula of the second author [10] in our proof of part (3) of the Main Theorem. The proof of this result is combinatorial and relies on the enumeration of words in certain monoids. with type (r 1 , . . . , r n ) and d connected components is When k < d, it is clear that there are no partitions of [n] of type (r 1 , . . . , r n ) with i r i = k and d connected components; there is a unique (nonnesting) partition of [n] with type (n, 0, . . . , 0) and it has n connected components.
We remark that the product formula in Lemma 2.3 was predicted from the formula (5.1) for Ish arrangements, not by studying nonnesting partitions directly. This is one case in which the Ish arrangement acted as a "toy model" for other Catalan objects.

Characteristic Polynomials
In this section we explicitly compute the characteristic polynomials of Shi(G) and Ish(G) and observe that they are equal. The formula is expressed in terms of G-Stirling numbers Stir(G, k). Our tools are the finite field method of Crapo and Rota and the principle of inclusion-exclusion. Zaslavsky's theorem then implies that Shi(G) and Ish(G) have the same number of regions and the same number of relatively bounded regions (regions with one degree of freedom).
3.1. The method. Let A be a finite hyperplane arrangement in R n and suppose that the defining equations for hyperplanes in A have coefficients in Z. Then the finite field method of Crapo and Rota [5] is a useful way to compute the characteristic polynomial of A without having to know its intersection poset. Let p ∈ Z be prime and consider a hyperplane H ⊆ R n with fixed defining equation a 1 x 1 + · · · + a n x n = b, where a i , b ∈ Z. Then we define the following subset H p of the finite vector space F n p by reducing the coefficients of H modulo p: . , x n ) ∈ F n p : a 1 x 1 + · · · + a n x n = b}. Observe that H p may not be a hyperplane in F n p when p is small, and that H p in general depends on the defining equation chosen. However: If p is large enough then each H p is a hyperplane in F n p and the characteristic polynomial of A has a nice relationship to the reduced hyperplane arrangement A p := {H p : H ∈ A} in F n p . Theorem 3.1. [5] Let p ∈ Z be a large prime, and let A be a finite collection of hyperplanes in R n whose hyperplanes have defining equations with coefficients in Z. Then the characteristic polynomial of A satisfies counts the number of points in the complement of the reduced arrangement A p in the finite vector space F n p .
3.2. The calculation. Now we use the finite field method to compute the characteristic polynomials of Shi(G) and Ish(G). We observe that they are equal.
2 be a graph on n vertices. The characteristic polynomials of the deleted Shi and Ish arrangement are given by: Proof. Let p ∈ Z be a large prime. We will show that the reduced complements F n p − Shi(G) p and F n p − Ish(G) p (forgive the abuse of notation) contain the same number of points, counted by the above formula.
To do this, we identify {0, 1, . . . , p − 1} = F p with the vertices of a regular p- The complements of Shi(G) p and Ish(G) p are both contained in F n p − Cox(n) p , so we must count certain kinds of injective labelings.
First we deal with Shi(G) p . For any set of edges S ⊆ G let f (S) denote the number of vectors v ∈ F n p − Shi(G) p such that v i − v j = 1 for all edges ij ∈ S (this notation implies i < j). By the principle of inclusion-exclusion (see for example [12,Chapter 2]) we observe that the number of points in F n p − Shi(G) p is equal to Now suppose that S contains edges ij and iℓ with the same left endpoint. The Similarly f (S) = 0 whenever S contains two edges with the same right endpoint. That is, the sets S that contribute to the sum (3.1) are precisely the arc sets of G-partitions.
Let S ⊆ G correspond to a G-partition with n − k blocks (that is, |S| = k). To compute f (S) note that the conditions v i − v j = 1 for all ij ∈ S imply that the p-gon F p gets labeled by n − k − 1 (given) contiguous strings of labels with spaces between. There are (n − k − 1)! ways to cyclically permute the strings; there are p−k−1 n−k−1 ways to place p − n empty spaces between the strings; and there are p ways to choose the "origin" (the location of 0). Hence: Combining (3.1) and (3.2) with the finite field method gives the desired formula.
We use a parallel argument to deal with Ish(G) p . For any set of edges S ⊆ G let g(S) be the number of vectors v ∈ F n p − Ish(G) p such that v 1 − v j = i for all ij ∈ S. As above, the points of F p − Ish(G) p are counted by and one can check that g(S) = 0 unless S is the arc set of a G-partition. We let S correspond to a G-partition with n − k blocks (that is, |S| = k) and compute g(S) as follows. First choose v 1 in p ways. Then for each ij ∈ S the condition v 1 − v j = i uniquely determines the value of v j . The remaining n − k − 1 labels must be placed injectively in the remaining p − k − 1 positions and there are (p − k − 1)(p − k − 2) · · · (p − n + 1) ways to do this. Thus we get the desired formula: Notice that the counting argument for computing χ Ish(G) (p) was more straightforward than the argument for χ Shi(G) (p). This again agrees with our observation that Ish is a toy model for Shi. It is somewhat surprising that the two inclusion-exclusion arguments result in the same expression. It may be interesting to find a direct bijection between the points of the complements F n p − Shi(G) p and F n p − Ish(G) p .

Remarks.
A simplified version of the above argument shows that the characteristic polynomials of Shi(n) and Ish(n) (the case of the complete graph) are both equal to p (p − n) n−1 . This result was obtained earlier by Headley [6] and Athanasiadis [3] (for the Shi arrangement) and by the first author [ has the following property: if i < j < k and ij ∈ G, then ik ∈ G. Then we have where d i := #{j : ij ∈ G} is the outdegree of vertex i in G.
In the same paper, Athanasiadis showed that the arrangements Shi(G) of the Corollary are free in the sense of Terao [13] (see also [8]). This is an open problem for the corresponding Ish arrangements Ish(G).
We remark that the characteristic polynomial of an arrangement allows us to count certain kinds of regions. Some notation: Let A be a finite collection of hyperplanes in R n and suppose that the normals to the hyperplanes span a space V ⊆ R n of dimension r. This r is called the rank of the arrangement. If r < n then the arrangement A has no bounded regions; in this case we say that a region of A is relatively bounded if its intersection with V is bounded. The following is a classic theorem of Zaslavsky. Observe that the normals to either Shi(G) or Ish(G) span the hyperplane x 1 + x 2 + · · · + x n = 0. Hence each of these arrangements has rank n − 1. It follows that neither arrangement has bounded regions and its relatively bounded regions have one degree of freedom. In the case of the complete graph, we find that the arrangements Shi(n) and Ish(n) both have (n + 1) n−1 regions and (n − 1) n−1 regions with one degree of freedom.
The fact that the Shi arrangement Shi(n) has (n + 1) n−1 regions was first proved by Jian-Yi Shi (see [11]). This beautiful result has motivated more than a few research papers since 1985 (including the present one).

Fix a graph G ⊆ [n]
2 . In this section we devise combinatorial labels for the regions of the deleted arrangements Shi(G) and Ish(G); we call these labels Shi ceiling diagrams and Ish ceiling diagrams, respectively. (Something like "Shi floor diagrams" appeared earlier in Athanasiadis and Linusson [2].) Essentially, each diagram encodes the ceilings of a given region, from which we can easily determine its recession cone.

Shi ceiling diagrams.
Recall that the regions (cones) of the Coxeter arrangement Cox(n) correspond to elements of the symmetric group S(n). If C ⊆ R n is the dominant cone -defined by the coordinate inequalities x 1 > x 2 > · · · > x n -then the collection of regions of Cox(n) is {wC : w ∈ S(n)}, where wC is defined by the coordinate inequalities (4.1) x w(1) > x w(2) > · · · > x w(n) .

Now let R be a region of the deleted Shi arrangement Shi(G).
Since Cox(n) ⊆ Shi(G), R is contained in some cone wC. In this case, what are the possible ceilings of R? We note that the hyperplanes of Shi(G) that intersect wC are precisely (We can think of these as the non-inversions of w contained in G.) Furthermore, suppose that the region R is "below" some hyperplane x w(i) − x w(j) = 1 -that is, suppose that each v ∈ R satisfies v w(i) − v w(j) < i. Then, considering (4.1), R is also below any hyperplane of the form x w(i ′ ) − x w(j ′ ) = 1 such that That is, if we declare a partial order on Φ + (G, w) by saying that x w(i ′ ) − x w(j ′ ) = 1 is "less than" x w(i) − x w(j) = 1 when condition (4.2) holds, then the collection of Shi(G)-hyperplanes above R forms a down-closed set.
Theorem 4.1. There is a bijection between regions of Shi(G) in the cone wC and order ideals (down-closed sets) in the poset Φ + (G, w). This map sends a region R to the set of hyperplanes in Shi(G) that are "above" R (contain R and the origin in the same half space). The maximal elements of the ideal are the ceilings of R.
Proof. Let R be a region of Shi(G) contained in wC. We showed above that the collection of hyperplanes above R is an order ideal in Φ + (G, w). The map is injective since these hyperplanes uniquely determine R. Observe that the ceilings of R are the elements of the ideal that may be individually removed to obtain another ideal, and these are precisely the maximal elements. We refer to Athanasiadis and Linusson [2] for the proof that every ideal corresponds to a non-empty region. To express this combinatorially, we note that order ideals in Φ + (G, w) are equivalent to nonnesting G-partitions whose blocks are "increasing" with respect to w. Indeed, there is a bijection between ideals and antichains (sets of pairwise-incomparable elements), since an ideal is uniquely determined by its antichain of maximal elements. By sending the hyperplane x w(i) − x w(j) = 1 to the arc ij, each antichain in Φ + (G, w) corresponds to a G-partition of [n] whose arcs i < j satisfy w(i) < w(j). Finally, note that two arcs nest if and only if they are comparable in the poset Φ + (G, w).
Following these remarks, we draw a diagram for each region of Shi(G).
Definition 4.1. Let R be a region of Shi(G) contained in the cone wC. We associate R with the pair (w, π) where π is an order ideal in the poset Φ + (G, w) of non-inversions of w contained in G. Equivalently, π is a nonnesting G-partition whose blocks are increasing with respect to w. We draw (w, π) by placing the arc diagram for π above the numbers w(1), . . . , w(n), and we call this the Shi ceiling diagram of R.
. Moreover, if ij ∈ G with i and j in the same block of π then the coordinate inequality x w(i) − x w(j) < 1 holds on R and we must have v w(i) = v w(i+1) = · · · = v w(j) . Since these are the only constraints on v, we conclude that the recession cone Rec(R) consists of all vectors of the form w · (a 1 , a 2 , . . . , a n ), where a 1 ≥ · · · ≥ a n and where a i = a j if i and j are in the same connected component of π. The dimension of the cone is therefore d.  form (a, a, b, b, a, a, b, a) ∈ R 8 with a ≥ b, and it has dimension 2.

4.2.
Ish ceiling diagrams. In order to compare the two arrangements, we now define an Ish analogue of Shi ceiling diagrams.
Since Cox(n) ⊆ Ish(G), each region R of Ish(G) is contained in wC for some permutation w ∈ S(n), in which case each vector v = (v 1 , . . . , v n ) ∈ R satisfies Which Ish(G)-hyperplanes are the possible ceilings of this region? If the hyperplane x 1 − x j = i intersects the cone wC it must be true that x 1 > x j on wC (since i is positive). Considering (4.3), this means that j must occur to the right of 1 in the list w(1), . . . , w(n) -that is, we must have w −1 (1) < w −1 (j). We conclude that the Ish hyperplanes that intersect the cone wC are precisely Now let R be a region of Ish(G) in the cone wC and suppose that R is below Then it is easy to check that R is also below the hyperplane By analogy with the Shi case, we define a partial order on Ψ + (G, w) by declaring that the hyperplane x 1 − x j = i is "less than" the hyperplane x 1 − x j ′ = i ′ whenever (4.4) holds. This leads to a useful characterization of Ish(G) regions.

Theorem 4.3.
There is a bijection between regions of Ish(G) in the cone wC and order filters (up-closed sets) in the poset Ψ + (G, w). This map sends a region R to the set of hyperplanes in Ish(G) that are "above" R (contain R and the origin in the same half space). The minimal elements of the filter are the ceilings of R.
Proof. Let R be a region of Ish(G) in the cone wC. By the above remarks we know that the collection of Ish(G)-hyperplanes above R is an order filter in Ψ + (G, w). These hyperplanes together with the fact that R lies in wC uniquely determine R, so the map is injective. The ceilings of R are precisely the elements of this filter that may be individually removed to obtain another filter -that is, they are the minimal elements.
To show that the map is surjective, we must show that each filter in Ψ + (G, w) corresponds to a non-empty region of Ish(G). Let F ⊆ Ψ + (G, w) be an order filter and let A ⊆ F be its set of minimal elements. For 1 ≤ i ≤ n define where we adopt the convention that max(∅) = 0. One may check that the point (z w(1) , . . . , z w(n) ) ∈ R n lies on the boundary of a region of Ish(G) which maps to the filter F . (Alternatively, note that Theorems 3.2 and 5.1 imply, respectively, that the Stir(G, n − k) n! (k + 1)! .
Hence any injective map between then must be surjective.) It is convenient to express this situation with a picture. Given w ∈ S(n) we draw w(1), w(2), . . . , w(n) on a line. For each j to the right of 1 we draw j − 1 boxes above the symbol j. If we identify the ith box above j with the hyperplane x 1 − x j = i then the collection of boxes is exactly Ψ + (G, w) (we erase the boxes that are not in G); the partial order on boxes increases up and to the left.
In Figure 4.3 (left side) we have drawn the poset Ψ + (G, w) for the complete graph G = K 8 and the permutation w = 51286347 ∈ S(8). The circles (closed up and to the left) indicate an order filter in this poset. This filter defines a region R of Ish(K 8 ) = Ish(8) in the cone wC and its ceilings are the antichain of minimal elements (hollow circles): To simplify the diagram further (right side), we just draw i hollow circles above the symbol j for each ceiling x 1 − x j = i. This is the Ish ceiling diagram of the region. We will encode it with the pair (w, ε) where ε i is the number of circles above the symbol w(i). For the example in Figure 4.3 we have (w, ε) = (51286347, (0, 0, 0, 1, 0, 0, 3, 5)). In order to count regions later, here is a purely combinatorial characterization of Ish ceiling diagrams.
2 be a graph and consider a permutation w ∈ S(n). We call the pair (w, ε) an Ish ceiling diagram if the vector ε = (ε 1 , . . . , ε n ) satisfies: • the nonzero entries of ε strictly increase.
We will draw the pair (w, ε) by placing w(1), . . . , w(n) on a line and drawing ε i circles above w(i). By the above remarks, the pair (w, ε) corresponds to a unique region of Ish(G) with a ceiling x 1 − x w(i) = i for each ε i = 0.
Finally, we can read the recession cone of a region directly from its Ish ceiling diagram.
Lemma 4.4. Let R be a region of Ish(G) in the cone wC with ceiling diagram (w, ε). If k is the maximum index such that ε k = 0 (or k = w −1 (1) if ε is the zero vector) then R has n − k + w −1 (1) degrees of freedom. In particular, the region R is relatively bounded (has 1 degree of freedom) if and only if w(1) = 1 and ε n = 0.
For example, consider the Ish ceiling diagram (w, ε) in Figure 4.3 and the corresponding region R of Ish (8). In this case we have w(1) = 5 and k = 8 is the largest index such that ε k = 0. Hence the recession cone Rec(R) consists of all vectors (a, a, a, a, b, a, a, a) ∈ R 8 with a ≥ b, and it has dimension 2.  Proof. This is essentially a picture proof. For the identity permutation w = 1 we observe that the posets Φ + (G, 1) and Ψ + (G, 1) look exactly the same, except that one is reflected in a line of slope 1. For example, here are the posets corresponding to the graph G = However, we gain something by summing over the cones wC. Not only do Shi(G) and Ish(G) have the same number of (unrestricted) regions with c ceilings, they have the same number of regions with c ceilings and d degrees of freedom. We prove this in the next section using a non-bijective method.

Counting the regions
In this section we introduce a partition-valued statistic on the regions of Shi(G) and Ish(G), and in each case we call this the ceiling partition of the region. (This concept is new even for the Shi arrangement.) It turns out that Shi(G) and Ish(G) have the same number of regions R with a given ceiling partition π; moreover, when the partition π has k blocks (i.e. R has n − k ceilings), this number has a beautiful formula: n!/(n − k + 1)!. The partition π does not determine the degrees of freedom of R. However, we still have a nice formula: The number of regions of Shi(G) or Ish(G) with ceiling partition π (with k blocks) and d degrees of freedom equals This completes the proof of the Main Theorem. At the end we make comments and suggestions for future research.

Ceiling partitions.
To each region R of Shi(G) or Ish(G) we associate a partition of the set [n], called its ceiling partition. We note that this partition may be nesting, and in general every partition of [n] will occur. The ceiling partition is determined by the ceilings of R and the cone wC in which R occurs; thus we can read it from the ceiling diagram. We will see that the correct language for ceiling partitions is the endpoint notation (a, b), discussed in Section 2.
(1) Let R be a region of Shi(G) with ceiling diagram (w, π). Then the ceiling partition of R is w · π (w acting on π). That is, the ceiling partition has w(i) and w(j) in a block whenever i and j are in a block of π. For example, the region in (2) Let R be a region of Ish(G) with ceiling diagram (w, ε). We define a pair of vectors (a, b) such that a i is the ith nonzero entry of ε, which occurs in position w −1 (b i ). The conditions of Definition 4.2 guarantee that (a, b) is the endpoint notation for a partition, which we call the ceiling partition of R. For example, the region shown in Figure 4.3 has ceiling partition (a, b) = (135, 847) since there is one circle above 8, three above 4, and five above 7. Again, the ceiling partition has c arcs if and only if R has c ceilings. (2) The number of regions of A with ceiling partition (a, b) and d degrees of freedom is To obtain the number of regions with c ceilings and d degrees of freedom, sum (5.1) over G-partitions (a, b) with k = n − c blocks.
Proof. First we deal with A = Shi(G). Recall that a Shi ceiling diagram (w, π) is a nonnesting partition π whose blocks are increasing with respect to the permutation w. Thus, to create a ceiling diagram (region) with ceiling partition (a, b), we must first choose a nonnesting partition π 0 with the same block sizes as (a, b) and then put the labels from each block of (a, b) (increasingly) in a block of π 0 . So suppose that (a, b) has r i blocks of size i. By Lemma 2.1 there are n! (n − k + 1)!r 1 !r 2 ! · · · r n ! ways to choose π 0 . Then, there are r 1 !r 2 ! · · · r n ! ways to map each block of (a, b) to a block of π 0 with the same size. This proves (1). To prove (2), note that the region (w, π) has d degrees of freedom if and only if the nonnesting partition π has d connected components. Apply the same argument as above, but use Lemma 2.2.
Next we deal with A = Ish(G). We wish to create an Ish region (w, ε) with ceiling partition (a, b). To do this, we choose w(1), . . . , w(n) and then place a i circles above the symbol b i . This will be an Ish ceiling diagram as long as the symbols b i occur in order, to the right of 1. That is, the permutation w must satisfy (5.2) w −1 (1) < w −1 (b 1 ) < · · · < w −1 (b n−k ).
There are n n−k+1 ways to place these symbols and then (k − 1)! ways to place the remaining symbols, proving (1). To prove (2), recall that (w, ε) has n − j + w −1 (1) degrees of freedom, where j is the largest index such that ε j = 0. In our case j = w −1 (b n−k ), so w must satisfy the condition w −1 (b n−k ) − w −1 (1) = n − d. First we can choose the pair (w −1 (1), w −1 (b n−k )) in d ways. Having done this, the rest of the permutation is subject to (5.2). There are n−d−1 n−k−1 ways to place symbols b 1 , . . . , b n−k−1 (left to right) in the n − d − 1 positions between 1 and b n−k , and then there are (k − 1)! ways to place the remaining k − 1 symbols. The result follows.
Once again, note that this proof was more direct for Ish than for Shi. In fact, the calculation of formula (5.1) for Ish was the inspiration for Lemma 2.2.

5.3.
Concluding remarks. The notion of a ceiling partition has independent interest, beyond the proof of Theorem 5.1. In particular, it leads to a new proof that the Shi arrangement Shi(n) has (n + 1) n−1 regions. Consider the collection of maps from [n] into a set of size x. On one hand, there are x n of these. On the other hand, there are Stir(n, k)x(x − 1) · · · (x − k + 1) such maps with image of size k, where Stir(n, k) is the number of partitions of [n] into k blocks (fibers). This proves the famous polynomial identity: x n = n k=1 Stir(n, k)x(x − 1) · · · (x − k + 1).
Dividing by x and substituting x = n + 1 yields (n + 1) n−1 = n k=1 Stir(n, k) n! (n − k + 1)! , where the right hand side counts regions of Shi(n) by the number k of blocks in their ceiling partition.
Finally, here are some problems for future research.
(1) The original motivation for this paper was to find a bijection between regions of Shi and Ish. We solved this problem for dominant regions, but not in general. Based on Theorem 5.1, one should look for a bijection between Shi ceiling diagrams (w, π) and Ish ceiling diagrams (w, ε) with a fixed ceiling partition (a, b) and d degrees of freedom. Note that this bijection cannot preserve the permutation w.
(2) Following Theorem 3.2, find a direct bijection between points of the finite vector space F n p in the complements of the Shi arrangement Shi(G) p and the Ish arrangement Ish(G) p .
(3) The Shi arrangement is a famous example of a free hyperplane arrangement.
Investigate the freeness of Ish arrangements.

(5)
To what extent do the results of this paper apply to other deformations of the Coxeter arrangement?
(6) The deleted Shi arrangements exist for arbitrary crystallographic reflection groups. Define and study Ish arrangements for other reflection groups. Ish arrangements were invented to study q, t-Catalan numbers; this feature should extend to other types.