On floors and ceilings of the k-Catalan arrangement

The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $\Phi$ is a well-studied object enumerated by the Fu{\ss}-Catalan number $Cat^{(k)}(\Phi)$. It is natural to refine this enumeration by considering floors and ceilings of dominant regions. A conjecture of Armstrong states that counting dominant regions by their number of floors of a certain height gives the same distribution as counting dominant regions by their number of ceilings of the same height. We prove this conjecture using a bijection that provides even more refined enumerative information.


Introduction
Let Φ be a crystallographic root system of rank n with simple system S, positive system Φ + , and ambient vector space V . For background on root systems see [Hum90]. For k a positive integer, we define the k-Catalan arrangement of Φ as the hyperplane arrangement given by the hyperplanes H r α = {x ∈ V | x, α = r} for α ∈ Φ and r ∈ {0, 1, . . . , k}. The complement of this arrangement falls apart into connected components which we call the regions of the arrangement. Those regions R that have x, α > 0 for all α ∈ Φ + and all x ∈ R we call dominant. The number of dominant regions of the k-Catalan arrangement equals the Fuß-Catalan number Cat (k) (Φ) [Ath04] of Φ. This number remains somewhat mysterious, in the sense that it also counts other objects in combinatorics, like the set of k-divisible noncrossing partitions N C (k) (Φ) of Φ [Arm09, Theorem 3.5.3] and the number of facets of the k-generalised cluster complex ∆ (k) (Φ) of Φ [FR05,Proposition 8.4], but no uniform proof of this fact is known, that is every known proof of this fact appeals to the classification of irreducible crystallographic root systems.
For a dominant region R of the k-Catalan arrangement, we call those hyperplanes that support a facet of R the walls of R. Those walls of R which do not contain the origin and have the origin on the same side as R we call the ceilings of R. The walls of R that do not contain the origin and separate R from the origin are called its floors. We say a hyperplane is of height r if it is of the form H r α for α ∈ Φ + .
One reason why floors and ceilings of dominant regions are interesting is that they give a more refined enumeration of the dominant regions of the k-Catalan arrangement of Φ that corresponds to refined enumerations of other objects counted by the Fuß-Catalan number Cat For the special case where Φ is of type A n−1 , more is known. For example, there is an explicit bijection between the set of dominant regions of the k-Catalan arrangement of Φ and the set of facets of the cluster complex of Φ [FKT13]. There is also an enumeration of those dominant regions that have a fixed hyperplane as a floor [FTV13]. In contrast to those results, all results in this paper are stated and proven uniformly for all crystallographic root systems without appeal to the classification.  See Figure 1 for an example. From this theorem, we obtain some enumerative corollaries. In particular, let f l r (l) be the number of dominant regions in the k-Catalan arrangement that have exactly l floors of height r, and let cl r (l) be the number of dominant regions that have exactly l ceilings of height r [Arm09, Definition 5.1.23]. We deduce the following conjecture of Armstrong.
. We have f l r (l) = cl r (l) for all 1 ≤ r ≤ k and 0 ≤ l ≤ n.
Specialising to the k = 1 case, we also give a geometric interpretation in terms of dominant regions of the Catalan arrangement of the Panyushev complement on ideals in the root poset of Φ.

Definitions
For this section and the next one, suppose that Φ is irreducible. Define the affine Coxeter arrangement of Φ as the union of all hyperplanes of the form H r α = {x ∈ V | x, α = r} for α ∈ Φ and r ∈ Z. Then the complement of this falls apart into connected components, all of which are congruent open n-simplices, called alcoves. The affine Weyl group W a generated by all the reflections through hyperplanes of the form H r α for α ∈ Φ and r ∈ Z is a Coxeter group, with generating set S a = {s 0 , s 1 , . . . , s n }, where s 1 , . . . , s n are the reflections in the hyperplanes orthogonal to the simple roots of Φ and s 0 is the reflection in H 1 α , whereα is the highest root of Φ.
The group W a acts simply transitively on the alcoves, so if we define the fundamental alcove as then every alcove A can be written as w(A • ) for a unique w ∈ W a .
Clearly any alcove is contained in exactly one region R of the k-Catalan arrangement of Φ. For any alcove A in the affine Coxeter arrangement of Φ and α ∈ Φ + , there exists a unique integer r with r − 1 < x, α < r for all x ∈ A. We denote this integer by r(A, α).
Suppose that for each α ∈ Φ + we are given a positive integer r α . The following is due to Shi [Shi87, Theorem 5.2].
Define a partial order on Φ + by α ≤ β if and only if β − α ∈ S N , that is, β ≥ α if and only if β − α can be written as a linear combination of simple roots with nonnegative integer coefficients. The set of positive roots Φ + with this partial order is called the root poset. A subset I ⊆ Φ + is called an ideal if for all α ∈ I and β ≤ α, also β ∈ I. A subset J ⊆ Φ + is called an order filter if for all α ∈ J and β ≥ α, also β ∈ J.
Suppose I = (I 1 , I 2 , . . . , I k ) is an ascending (multi)chain of k ideals in the root poset of Φ, that is I 1 ⊆ I 2 ⊆ . . . ⊆ I k . Setting J i = Φ + \I i for i ∈ [k] and J = (J 1 , J 2 , . . . , J k ) gives us the corresponding descending chain of order filters. That is, we have J 1 ⊇ J 2 ⊇ . . . ⊇ J k . The ascending chain of ideals I and the corresponding descending chain of order filters J are both called geometric if the following conditions are satisfied simultaneously.
Let R be a dominant region of the k-Catalan arrangement of Φ. Define θ(R) = (I 1 , I 2 , . . . , I k ) and φ(R) = (J 1 , J 2 , . . . , J k ), where for i ∈ {0, 1, . . . , k}. It is not difficult to verify that θ(R) is a geometric chain of ideals and that φ(R) is the corresponding geometric chain of order filters.
For a geometric chain of order filters J = (J 1 , J 2 , . . . , J k ), and α ∈ Φ + , we define It turns out that φ is a bijection from the set of dominant regions of the k-Catalan arrangement of Φ to the set of geometric chains of k order filters in the root poset of Φ [Ath05, Theorem 3.6]. Its inverse ψ is the map sending a geometric chain of order filters J to the region R of the k-Catalan arrangement containing the alcove A with r(A, α) = k α (J ) + 1 for all α ∈ Φ + . This alcove A is called the minimal alcove of R. We call α ∈ Φ + a rank r indecomposable element [Ath05, Definition 3.8] of a geometric chain of order filters We have that H r α is a floor of R if and only if α is a rank r indecomposable element of the geometric chain of order filters J = φ(R) [Ath05, Theorem 3.11].
We will soon see that H r α is a ceiling of R if and only if α is a rank r indecomposable element of the geometric chain of ideals I = θ(R).

Lemmas
Our aim for this rather technical section is to prove the following theorem.
Theorem 3.1. Let R be a dominant region in the k-Catalan arrangement of Φ, I = θ(R) and α ∈ Φ + . Then R contains an alcove B such that for all r ∈ [k] the following are equivalent: (1) H r α is a ceiling of R, (2) α is a rank r indecomposable element of I, and (3) H r α is a ceiling of B. It is already known that Theorem 3.1 holds for bounded dominant regions [AT06,Theorem 3.11]. In that case, we may take the alcove B to be the maximal alcove of the bounded region R.
Our approach to proving Theorem 3.1 is to note that when a region R of the k-Catalan arrangement is subdivided into regions of the (k + 1)-Catalan arrangement by hyperplanes of the form H k+1 α for α ∈ Φ + , at least one of the resulting regions is bounded. We find a region R of the (k + 1)-Catalan arrangement which, among the bounded regions of the (k + 1)-Catalan arrangement that are contained in R, is the one furthest away from the origin. We call the maximal alcove B of R the pseudomaximal alcove of R. It equals the maximal alcove of R if R is bounded. The alcove B ⊆ R will be seen to satisfy the assertion of Theorem 3.1. Instead of working directly with the dominant regions of the k-and (k + 1)-Catalan arrangements, we usually phrase our results in terms of the corresponding geometric chains of ideals. We require the following lemmas: Lemma 3.4 ([AT06, Lemma 3.10]). Suppose α is an indecomposable element of I.
We are now ready to define the bounded dominant region R of the (k+1)-Catalan arrangement in terms of the corresponding geometric chain of k + 1 ideals I. For a geometric chain of ideals I = (I 1 , I 2 , . . . , I k ), let I i = I i for all i ∈ [k] and let I k+1 = i+j=k+1 ((I i + I j ) ∩ Φ + ) ∪ I k ∪ S. By Lemma 3.5, I k+1 is an ideal. Define I = (I 1 , . . . , I k+1 ). Proof. By construction, I is an ascending chain of ideals. If i + j ≤ k, we have that ( Let J = (J 1 , J 2 , . . . , J k ) be the geometric chain of order filters corresponding to the geometric chain of ideals I. Define J similarly. We need to verify that Suppose next that i + j = k + 1. Take any region R of the (k + 1)-Catalan arrangement that is contained in R. Let θ(R ) = I = (I 1 , I 2 , . . . , I k+1 ) be the geometric chain of ideals corresponding to R and let J = (J 1 , J 2 , . . . , J k+1 ) be the corresponding geometric chain of order filters. Then R and R are on the same side of each hyperplane of the k-Catalan arrangement. Thus I l = I l and J l = J l for l ∈ [k]. Thus we have The sum of two positive roots is never a simple root, so we even have ( Thus the chain of ideals I is geometric. It is also clearly positive, so R = θ −1 (I) is bounded. Since I i = I i for i ∈ [k], R and R are on the same side of each hyperplane of the k-Catalan arrangement, so R is contained in R.  Proof. First note that α ∈ supp(I) N implies that r α (I) < ∞. So may write α = α 1 + α 2 + . . . + α m with α i ∈ I ri for i ∈ [m] and r 1 + r 2 + . . . + r m = r α (I).
Since α i ∈ I ri = I ri this implies that r α (I) ≤ r α (I).
For R a dominant region of the k-Catalan arrangement, define the pseudomaximal alcove of R to be the maximal alcove of R. This term is justified by the following proposition.
Proposition 3.8. If R is a bounded dominant region of the k-Catalan arrangement, its pseudomaximal alcove is equal to its maximal alcove.
Proof. Let A and B be the maximal and pseudomaximal alcoves of R respectively. If I = θ(R), then r(α, A) = r α (I) for all α ∈ Φ + . Since B is the maximal alcove of R, we have r(α, B) = r α (I) for all α ∈ Φ + . Now I is positive since R is bounded, so supp(I) = S. Thus r α (I) = r α (I) for all α ∈ Φ + by Lemma 3.7. So r(α, A) = r(α, B) for all α ∈ Φ + and therefore A = B.
Lemma 3.9. Let R be a region of the k-Catalan arrangement of Φ, let be B be its pseudomaximal alcove and let t ≤ k be a positive integer. If x 0 , α > t for some x 0 ∈ R, then x, α > t for all x ∈ B.
Lemma 3.10. If α is a rank r indecomposable element of I, then α is a rank r indecomposable element of I.
Lemma 3.11. If α ∈ Φ + and H r α is a ceiling of a dominant region R of the k-Catalan arrangement, then α is a rank r indecomposable element of I = θ(R).
Proof. Since the origin and R are on the same side of H r α , we have that x, α < r for all x ∈ R, so α ∈ I r and thus r α (I) ≤ r. But if r α (I) = i < r, then α ∈ I i by Lemma 3.3, so x, α < i ≤ r − 1 for all x ∈ R. So H r α is not a wall of R, a contradiction. Thus r α (I) = r.
If α = β + γ for β ∈ I i and γ ∈ I j with i + j = r, then the fact that x, α < r for all x ∈ R is a consequence of x, β < i and x, γ < j for all x ∈ R, so H r α does not support a facet of R. So α / ∈ I i + I j for i + j = r.
If r α+β (I) = t ≤ k, then α + β ∈ I t by Lemma 3.3, so x, α + β < t for all x in R. If also x, β > t − r for all x ∈ R, then x, α < r for all x ∈ R is a consequence of these, so H r α does not support a facet of R. So x, β < t − r for all x ∈ R, so β ∈ I t−r .
Thus α is a rank r indecomposable element of I.
Proof of Theorem 3.1. We take B to be the pseudomaximal alcove of R, that is the maximal alcove of R. We will show that (1) ⇒ (2) ⇒ (3) ⇒ (1).
The statement that (1) ⇒ (2) is Lemma 3.11. For (2) ⇒ (3), suppose α is a rank r indecomposable element of I. Then by Lemma 3.10, α is also a rank r indecomposable element of I. So by Lemma 3.4, we have r α (I) = r β (I) + r γ (I) − 1 if α = β + γ for β, γ ∈ Φ + , and also r α (I) + r β (I) = r α+β (I) if β, α + β ∈ Φ + . Thus there exists an alcove B with r(B , β) = r β (I) for β = α and r(B , α) = r α (I) + 1 by Lemma 2.1. Since r(B, β) = r β (I) for all β ∈ Φ + , this means that B and B are on the same side of each hyperplane of the affine Coxeter arrangement, except for H rα(I) α = H r α . Thus H r α is a wall of B. Since H r α does not separate B from the origin, it is a ceiling of B.
For (3) ⇒ (1), suppose H r α is a ceiling of B. Let B be the alcove which is the reflection of B in the hyperplane H r α . Then x, α > r for all x ∈ B , so by Lemma 3.9 the alcove B is not contained in R. Thus H r α is a wall of R. It does not separate R from the origin, so it is a ceiling of R. This completes the proof.

Proof of Theorem 1.1
We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1. Let us at first suppose that Φ is an irreducible crystallographic root system of rank n. For m = 0, the statement is immediate. Suppose that 0 < m ≤ n.
To define the bijection Θ, let R ∈ U (M ) and let A be the minimal alcove of R. The reflections s i1 α1 , . . . , s im αm in the hyperplanes H i1 α1 , . . . , H im αm are reflections in facets of the alcove A = w(A • ), so the set S = {s i1 α1 , . . . , s im αm } equals wJw −1 for some J ⊂ S a and w ∈ W a . Thus the reflection group W generated by S is a proper parabolic subgroup of W a . In particular, it is finite. With respect to the finite reflection group W , the alcove A is contained in the dominant Weyl chamber, that is the set So if w 0 is the longest element of W with respect to the generating set S , the alcove A = w 0 (A) is contained in the Weyl chamber Proof. The origin is contained in the Weyl chamber w 0 (C) of W . Thus no reflection in W fixes the origin. We can write A = w 0 (A) as t r · · · t 1 (A) where t i ∈ W is a reflection in a facet of t i−1 · · · t 1 (A) for all i ∈ [r]. In fact, if w 0 = s 1 · · · s r with s i ∈ S for all i ∈ [r] is a reduced expression for w 0 in W , we can take t i = s 1 · · · s i−1 s i s i−1 · · · s 1 . So t i · · · t 1 (A) and t i−1 · · · t 1 (A) are on the same side of every hyperplane in the affine Coxeter arrangement of Φ except for the reflecting hyperplane of t i . Since t i does not fix the origin, if t i−1 · · · t 1 (A) is dominant, then so is t i · · · t 1 (A). Thus by induction on i, the alcove A is dominant, so R is dominant.
Consider the Coxeter arrangement of W , which is the hyperplane arrangement given by the reflecting hyperplanes of all the reflections in W . The action of W on V restricts to an action on the set of these hyperplanes. Since We show that Θ is a bijection by exhibiting its inverse Ψ, a map from L(M ) to U (M ). Suppose R ∈ L(M ). Let B be the alcove in R given by Theorem 3.1. Let R be the region that contains B = w 0 (B). Similarly to the proof of Claim 1, we have that R ∈ U (M ). So let Ψ(R ) = R .
Claim 2. The maps Θ and Ψ are inverse to each other, so Θ is a bijection.
Proof. Suppose R ∈ U (M ), R = Θ(R) and R = Ψ(R ). Use the same notation as above for the alcoves A, A , B and B . Suppose for contradiction that R = R. Then there is a hyperplane H = H r α of the k-Catalan arrangement that separates R and R . So H separates A and B . Now A and B are in the dominant Weyl chamber of W , so they are on the same side of each reflecting hyperplane of W . Thus H is not a reflecting hyperplane of W . Now we may write A as t r · · · t 1 (A), where t i ∈ W is a reflection in a facet of t i−1 · · · t 1 (A) for all i ∈ [r]. So t i · · · t 1 (A) and t i−1 · · · t 1 (A) are on the same side of every hyperplane in the affine Coxeter arrangement, except for the reflecting hyperplane of t i , which cannot be H. Thus by induction on i, the alcove A is on the same side of H as A. Similarly B is on the same side of H as B . So A and B are on different sides of H, a contradiction, as they are contained in the same region, namely R . Thus Ψ(Θ(R)) = R = R, so Ψ • Θ = id. Similarly Θ • Ψ = id, so Θ and Ψ are inverse to each other, so Θ is a bijection.
For any dominant alcove, at least one of its n + 1 facets must either be a floor or contain the origin, and at least one must be a ceiling. So it has at most n ceilings and at most n floors. So any dominant region R of the k-Catalan arrangement has at most n ceilings and at most n floors. Thus if m > n, both U (M ) and L(M ) are empty. This completes the proof in the case where Φ is irreducible.
Then the regions of the k-Catalan arrangement of Φ are precisely the sets of the form R 1 ⊕ R 2 where R i is a region of the k-Catalan arrangement of Φ i for i = 1, 2. The region R 1 ⊕ R 2 is dominant if and only if R 1 and R 2 are both dominant. A hyperplane H r α is a floor of R 1 ⊕ R 2 if and only if H r α is a floor of R i for some i = 1, 2. The same holds for ceilings. Say M = M 1 M 2 with H ij αj ∈ M i if α j ∈ Φ i for j ∈ [m] and i = 1, 2. Assume the theorem holds for Φ 1 and Φ 2 , giving us bijections Θ 1 and Θ 2 for Φ 1 together with M 1 and Φ 2 together with M 2 respectively. Then Θ(R 1 ⊕ R 2 ) = Θ 1 (R 1 ) ⊕ Θ 2 (R 2 ) gives the required bijection for Φ together with M . This completes the proof by induction on the number of irreducible components of Φ.

Corollaries
We deduce some enumerative corollaries of Theorem 1.1. For any set M of hyperplanes of the k-Catalan arrangement, let U = (M ) be the set of dominant regions R of the k-Catalan arrangement such that the floors of R are exactly the hyperplanes in M , and let L = (M ) be the set of dominant regions R of the k-Catalan arrangement such that the ceilings of R are exactly the hyperplanes in M . Proof. This follows from Theorem 1.1 by an application of the Principle of Inclusion and Exclusion.
Corollary 5.2. For any tuple (a 1 , a 2 , . . . , a k ) of nonnegative integers, the number of dominant regions R that have exactly a j floors of height j for all j ∈ [k] is the same as the number of dominant regions R that have exactly a j ceilings of height j for all j ∈ [k].
Proof. Sum Corollary 5.1 over all sets M containing exactly a j hyperplanes of height j for all j ∈ [k].
Proof of Corollary 1.2. Set a r = l and sum Corollary 5.2 over all choices of a j for all j = r.

The Panyushev complement
In the special case where k = 1, a geometric chain of ideals I is simply the single ideal I 1 , similarly a geometric chain of order filters J is just the single order filter J 1 . The indecomposable elements of an ideal I are then just its maximal elements [AT06, Lemma 3.9]. The indecomposable elements of an order filter J are just its minimal elements [Ath05, Lemma 3.9] [Thi14, Lemma 1].
There is a natural bijection between ideals and antichains of any poset that sends an ideal to the set of its maximal elements. Similarly, there is a natural bijection between order filters and antichains that sends an order filter to the set of its minimal elements.
So for an ideal I in the root poset of Φ, we define the Panyushev complement Pan(I) as the ideal generated by the minimal elements of the order filter J = Φ + \I. From the above considerations, this is a bijection from the set of order ideals of the root poset of Φ to itself.
For a region R of the Catalan arrangement, let α is a ceiling of R} and F L(R) = {α ∈ Φ + | H 1 α is a floor of R}. Since a region R in the Catalan arrangement corresponds to a unique ideal I = θ(R), which corresponds uniquely to the set of its maximal elements, which equals CL(R) by Theorem 3.1, the map CL : R → CL(R) gives a bijection from the set of dominant regions in the Catalan arrangement to the set of antichains in the root poset. That the same holds for the map F L : R → F L(R) follows from an analogous argument that can already be deduced from [Ath05, Theorem 3.11]. Theorem 6.1. For an ideal I in the root poset of Φ, the region θ −1 (Pan(I)) is the unique region of the Catalan arrangement of Φ whose ceilings are exactly the floors of the region θ −1 (I).
Proof. The set CL(θ −1 (Pan(I))) is the set of maximal elements of Pan(I), which equals the set of minimal elements of J = Φ + \I, which equals F L(θ −1 (I)). Since CL is a bijection, θ −1 (Pan(I)) is the only region R with CL(R ) = F L(θ −1 (I)).
We could rephrase Theorem 6.1 as Pan = θ • CL −1 • F L • θ −1 . The fact that the Panyushev complement has a natural interpretation in terms of the dominant regions of the Catalan arrangement may serve to explain why it seems to be of particular interest for root posets.

Acknowledgements
I thank Myrto Kallipoliti, Henri Mühle, Vivien Ripoll and Eleni Tzanaki for helpful comments and suggestions. I also wish to express my gratitude to the anonymous referee for a very careful reading of the manuscript.