Removahedral congruences versus permutree congruences

The associahedron is classically constructed as a removahedron, i.e. by deleting inequalities in the facet description of the permutahedron. This removahedral construction extends to all permutreehedra (which interpolate between the permutahedron, the associahedron and the cube). Here, we investigate removahedra constructions for all quotientopes (which realize the lattice quotients of the weak order). On the one hand, we observe that the permutree fans are the only quotient fans realized by a removahedron. On the other hand, we show that any permutree fan can be realized by a removahedron constructed from any realization of the braid fan. Our results finally lead to a complete description of the type cone of the permutree fans.


Introduction
This paper deals with particular polytopal realizations of quotient fans of lattice congruences of the weak order. The prototypes of such polytopes are the classical permutahedron Perm n realizing the weak order on permutations and the classical associahedron Asso n realizing the Tamari lattice on binary trees. These two polytopes belong to the family of permutreehedra realizing the rotation lattice on permutrees, which play a fundamental role in this paper. Permutrees were introduced in [PP18] to generalize and interpolate between permutations and binary trees, and explain the combinatorial, geometric and algebraic similarities between them. They were inspired by Cambrian trees [CP17,LP18] which provide a combinatorial model to the type A Cambrian lattices of [Rea06]. As the classical construction of the associahedron due to [SS93,Lod04] and its generalization to all Cambrian associahedra by [HL07], all permutreehedra are obtained by deleting inequalities in the facet description of the permutahedron Perm n . Such polytopes are called removahedra, and were studied in the context of graph associahedra in [Pil17].
In general, any lattice congruence ≡ of the weak order on S n defines a quotient fan F ≡ obtained by gluing together the chambers of the braid fan corresponding to permutations in the same congruence class [Rea05]. This quotient fan F ≡ was recently proven to be the normal fan of a polytope P ≡ called quotientope [PS19,PPR20]. As their normal fans all refine the braid fan, quotientopes belong to the class of deformed permutahedra studied in [Pos09,PRW08] (we prefer the name "deformed permutahedra" rather than "generalized permutahedra" as there are many generalizations of permutahedra). All deformed permutahedra are obtained from the permutahedron Perm n by moving facets without "passing a vertex" (in the sense of [Pos09]). Observe that not all deformed permutahedra are removahedra, since it is sometimes inevitable to move facets, not only to remove them.
The construction of [PP18] for permutreehedra and the construction of [PS19] for quotientopes seem quite different. The polytopes resulting from the former construction lie outside the permutahedron Perm n (outsidahedra) while those resulting from the latter construction lie inside the permutahedron Perm n (insidahedra). In fact, it was already observed in [PS19,Rem. 12] that the quotient fans of some lattice congruences cannot be realized by a removahedron. The first contribution of this paper is to show the following strong dichotomy between the lattices congruences of the weak order regarding realizability by removahedra.
Theorem 1. Let ≡ be a lattice congruence of the weak order on S n . Then (i) if ≡ is not a permutree congruence, then the quotient fan F ≡ is not the normal fan of a removahedron, (ii) if ≡ is a permutree congruence, then the quotient fan F ≡ is the normal fan of a polytope obtained by deleting inequalities in the facet description of any polytope realizing the braid fan (not only the classical permutahedron Perm n ).
This statement is based on the understanding of the inequalities governing the facet heights that ensure to obtain a polytopal realization of the quotient fan. These inequalities, given by pairs of adjacent cones of the fan and known as wall-crossing inequalities, define the space of all realizations of the fan. This space of realizations is an open polyhedral cone called type cone and studied by [McM73], and its closure is also known as the deformation cone [Pos09,PRW08]. For instance, the deformation cone of the permutahedron Perm n is the space of submodular functions, and corresponds to all deformed permutahedra [Pos09]. The main contribution of this paper is the facet description of the type cone of any permutree fan, thus providing a complete description of all polytopal realizations of the permutree fans. This description requires three steps: for the δ-permutree fan F δ corresponding to a decoration δ, we provide purely combinatorial descriptions (only in terms of δ) for • the subsets of [n] which correspond to rays of F δ (Proposition 32), • the pairs of subsets of [n] which correspond to exchangeables rays of F δ (Proposition 39), • the pairs of subsets of [n] which correspond to facets of the type cone of F δ (Proposition 45).
From this facet description, we derive summation formulas for the number of facets of the type cones of permutree fans, leading to a characterization of the permutree fans whose type cone is simplicial. As advocated in [PPPP19], this property is interesting because it leads on the one hand to a simple description of all polytopal realizations of the fan in the kinematic space [AHBHY18], and on the other hand to canonical Minkowski sum decompositions of these realizations. This paper opens the door to a description of the type cone of the quotient fan for any lattice congruence of the weak order on S n , not only for permutree congruences. Preliminary computations however indicate that the combinatorics of the facet description of the type cone of an arbitrary quotient fan is much more intricated than that of permutree fans.
The paper is organized as follows. Section 2 provides a recollection of all material needed in the paper, including polyhedral geometry and type cones (Section 2.1), lattice quotients of the weak order (Section 2.2), deformed permutahedra and removahedra (Section 2.3) and permutrees (Section 2.4). Section 3 is devoted to the proof of Theorem 1. Finally, the type cones of all permutree fans are described in Section 4.

Preliminaries
We start with preliminaries on polyhedral geometry, type cones, braid arrangements, quotient fans, shards, deformed permutahedra, removahedra and permutrees. The presentation is largely inspired by the papers [PPPP19, PS19, PP18] and we reproduce here some of their pictures.

Polyhedral geometry and type cones
We start with basic notions of polyhedral geometry (see G. Ziegler's textbook [Zie98]) and a short introduction to type cones (see the original work of P. McMullen [McM73] or their recent application to g-vector fans in [PPPP19]).

Polyhedral geometry
A hyperplane H ⊂ R n is a supporting hyperplane of a set X ⊂ R n if H ∩ X = ∅ and X is contained in one of the two closed half-spaces of R n defined by H.
We denote by R 0 R : = r∈R λ r r λ r ∈ R 0 the positive span of a set R of vectors of R n . A (polyhedral) cone is a subset of R n defined equivalently as the positive span of finitely many vectors or as the intersection of finitely many closed linear halfspaces. Its faces are its intersections with its supporting linear hyperplanes, and its rays (resp. facets) are its dimension 1 (resp. codimension 1) faces. A cone is simplicial if it is generated by a set of linearly independent vectors.
A (polyhedral) fan F is a collection of cones which are closed under faces (if C ∈ F and F is a face of C, then F ∈ F) and intersect properly (if C, D ∈ F, then C ∩ D is a face of both C and D). The chambers (resp. walls, resp. rays) of F are its codimension 0 (resp. codimension 1, resp. dimension 1) cones. The fan F is simplicial if all its cones are, complete if the union of its cones covers the ambient space R n , and essential if it contains the cone {0}. Note that every complete fan is the product of an essential fan with its lineality space (the largest linear subspace contained in all cones of F). Given two fans F, G in R n , we say that F refines G (and that G coarsens F) if every cone of G is a union of cones of F. In a simplicial fan, we say that two maximal cones are adjacent if they share a facet, and that two rays are exchangeable if they belong to two adjacent cones but not to their common facet.
A polytope is a subset of R n defined equivalently as the convex hull of finitely many points or as a bounded intersection of finitely many closed affine halfspaces. Its faces are its intersections with its supporting affine hyperplanes, and its vertices (resp. edges, resp. facets) are its dimension 0 (resp. dimension 1, codimension 1) faces. The normal cone of a face F of a polytope P is the cone generated by the outer normal vectors of the facets of P containing F . The normal fan of P is the fan formed by the normal cones of all faces of P .

Type cones
Fix an essential complete simplicial fan F in R n with N rays. Let G be the N × nmatrix whose rows are (representative vectors for) the rays of F. For any vector h ∈ R N , we define the polytope P h : = {x ∈ R n | Gx h}. In other words, P h has an inequality r | x h r for each ray r of F, where h r denotes the coordinate of h corresponding to r. Note that F is not necessarily the normal fan of P h . The vectors h for which this holds are characterized by the following classical statement. It already appeared in the study of Minkowski summands of polytopes [Mey74,McM73], and in the theory of secondary polytopes [GKZ08], see also [DRS10]. We present here a convenient formulation from [CFZ02, Lem. 2.1].
Proposition 2. Let F be an essential complete simplicial fan in R n and let G be the N × n-matrix whose rows are the rays of F. Then the following are equivalent for any vector h ∈ R N : 1. The fan F is the normal fan of the polytope P h : = {x ∈ R n | Gx h}.

For any two adjacent chambers
is the unique (up to rescaling) linear dependence with α, β > 0 between the rays of R ∪ S.
The inequalities in this statement are called wall-crossing inequalities. For convenience, let us denote by α R,S (t) the coefficient of t in the unique linear dependence between the rays of R ∪ S such that α R,S (r) + α R,S (s) = 2, so that the inequality above rewrites as t∈R∪S α R,S (t) h t > 0.
When considering the question of the realizability of a complete simplicial fan F by a polytope, it is natural to consider all possible realizations of this fan, as was done by P. McMullen in [McM73]. The type cone of F is the cone for any adjacent chambers R 0 R and R 0 S of F .
The type cone TC(F) is an open cone. We denote by TC(F) the closure of TC(F), and still call it the type cone of F. It is the closed polyhedral cone defined by the inequalities t∈R∪S α R,S (t) h t 0 for any adjacent chambers R 0 R and R 0 S. It describes all polytopes P h whose normal fans coarsen the fan F. If F is the normal fan of the polytope P , then TC(F) is also known as the deformation cone of P , see [Pos09,PRW08]. We use TC(F) rather than TC(F) when we want to speak about the facets of TC(F).
Also observe that the lineality space of the type cone TC(F) has dimension n (it is invariant by translation in GR n ). In particular, the type cone TC(F) is simplicial when it has N − n − 1 facets. While very particular, the fans for which the type cone is simplicial are very interesting as all their polytopal realizations can be described as follows.
Proposition 3 ([PPPP19, Coro. 1.11]). Let F be an essential complete simplicial fan in R n with N rays, such that the type cone TC(F) is simplicial. Let K be the (N −n)×Nmatrix whose rows are the inner normal vectors of the facets of TC(F). Then the polytope Q(u) : = z ∈ R N 0 Kz = u is a realization of the fan F for any positive vector u ∈ R N −n >0 . Moreover, the polytopes Q(u) for u ∈ R N −n >0 describe all polytopal realizations of F. the electronic journal of combinatorics 28(4) (2021), #P4.8 In this paper, we shall also use a non-simplicial version of Proposition 2, see [Mey74].
Proposition 4. Let F be an essential complete (non necessarily simplicial) fan in R n and G be the N ×n-matrix whose rows are the rays of F. Then the following are equivalent for any h ∈ R N : 1. The fan F is the normal fan of the polytope P h : = {x ∈ R n | Gx h}.
2. The coordinates of h satisfy the following equalities and inequalities: • for any chamber R 0 R of F and any linear dependence r∈R γ r r = 0 among the rays of R, we have r∈R γ r h r = 0, • for any two adjacent chambers R 0 R and R 0 S of F, any rays r ∈ R S and s ∈ S R, and any linear dependence α r + β s + t∈R∩S γ t t = 0 among the rays {r, s} ∪ (R ∩ S) with α, β > 0, we have α h r + β h s + t∈R∩S γ t h t > 0.

Geometry of lattice quotients of the weak order
We now recall the combinatorial and geometric toolbox to deal with lattice quotients of the weak order on S n . The presentation and pictures are borrowed from [PS19].

Weak order, braid fan, and permutahedron
Let S n denote the set of permutations of the set [n] : = {1, . . . , n}. The inversion set of a permutation σ ∈ S n is inv(σ) : = {(σ i , σ j ) | 1 i < j n and σ i > σ j }. We consider the weak order on S n defined by σ τ ⇐⇒ inv(σ) ⊆ inv(τ ). See Figure 1 (left). This order can be interpreted geometrically on the braid fan F n or the permutahedron Perm n defined below. The braid arrangement is the set H n of linear hyperplanes {x ∈ R n | x i = x j } for all 1 i < j n. As all hyperplanes of H n contain the line R1 : = R(1, 1, . . . , 1), we restrict  to the hyperplane H : = x ∈ R n i∈ [n] x i = 0 . The hyperplanes of H n divide H into chambers, which are the maximal cones of a complete simplicial fan F n , called braid fan.

It has
• a chamber C(σ) : = {x ∈ H | x σ 1 x σ 2 . . . x σn } for each permutation σ of S n , • a ray C(I) The permutahedron is the polytope Perm n defined equivalently as: • the convex hull of the points i∈[n] i e σ i for all permutations σ ∈ S n , • the intersection of the hyperplane The normal fan of the permutahedron Perm n is the braid fan F n . The Hasse diagram of the weak order on S n can be seen geometrically as the dual graph of the braid fan F n , or the graph of the permutahedron Perm n , when oriented in the linear direction α : = i∈[n] (2i − n − 1) e i = (−n + 1, −n + 3, . . . , n − 3, n − 1). See Figure 1.

Lattice congruences and quotient fans
A lattice congruence of a lattice (L, , ∧, ∨) is an equivalence relation on L that respects the meet and the join operations, i.e. such that x ≡ x and y ≡ y implies x ∧ y ≡ x ∧ y and x ∨ y ≡ x ∨ y . It defines a lattice quotient L/≡ on the congruence classes of ≡ where X Y if and only if there exist x ∈ X and y ∈ Y such that x y, and X ∧ Y (resp. X ∨ Y ) is the congruence class of x ∧ y (resp. x ∨ y) for any x ∈ X and y ∈ Y .
Example 5. The prototype lattice congruence of the weak order is the sylvester congruence ≡ sylv , see [LR98,HNT05]. Its congruence classes are the fibers of the binary search tree insertion algorithm, or equivalently the sets of linear extensions of binary trees (labeled in inorder and considered as posets oriented from bottom to top). It can also be seen as the transitive closure of the rewriting rule U ikV jW ≡ sylv U kiV jW for some letters i < j < k and words U, V, W on [n]. The quotient of the weak order by the sylvester congruence is (isomorphic to) the classical Tamari lattice [Tam51], whose elements are the binary trees on n nodes and whose cover relations are rotations in binary trees. The sylvester congruence and the Tamari lattice are illustrated in Figure 2  for n = 4. We will use the sylvester congruence and the Tamari lattice as a familiar example throughout the paper.
Lattice congruences naturally yield quotient fans, which turn out to be polytopal.
Theorem 6 ([Rea05]). Any lattice congruence ≡ of the weak order on S n defines a complete fan F ≡ , called quotient fan, whose chambers are obtained gluing together the chambers C(σ) of the braid fan F n corresponding to permutations σ that belong to the same congruence class of ≡.
Theorem 7 ([PS19]). For any lattice congruence ≡ of the weak order on S n , the quotient fan F ≡ is the normal fan of a polytope P ≡ , called quotientope.
By construction, the Hasse diagram of the quotient of the weak order by ≡ is given by the dual graph of the quotient fan F ≡ , or by the graph of the quotientope P ≡ , oriented in the direction α.
Example 8. For the sylvester congruence ≡ sylv of Example 5, the quotient fan F sylv has • a ray C(I) for each proper interval I = [i, j] [n].
Figures 3 and 4 (right) illustrate the quotient fans F sylv for n = 3 and n = 4. The quotient fan F sylv is the normal fan of the classical associahedron Asso n defined equivalently as: • the convex hull of the points j∈[n] (T, j) r(T, j) e j for all binary trees T on n nodes, where (T, j) and r(T, j) respectively denote the numbers of leaves in the left and right subtrees of the node j of T (labeled in inorder), see [Lod04], for all 1 i j n, see [SS93].

Shards
An alternative description of the quotient fan F ≡ defined in Theorem 6 is given by its walls, each of which can be seen as the union of some preserved walls of the braid arrangement. The conditions in the definition of lattice congruences impose strong constraints on the set of preserved walls: deleting some walls forces to delete others. Shards were introduced by N. Reading in [Rea03] (see also [Rea16b,Rea16a]) to understand the possible sets of preserved walls.
Throughout the paper, we use a convenient notation for shards borrowed from N. Reading's work on arc diagrams [Rea15]: we consider n dots on the horizontal axis, and we represent the shard Σ(i, j, S) by an arc joining the ith dot to the jth dot and passing above (resp. below) the kth dot when k ∈ S (resp. when k / ∈ S). For instance, the arc represents the shard Σ(1, 4, {2}). We say that Σ(i, j, S) is an up (resp. down) shard if the corresponding arc passes above (resp. below) all Figures 3 and 4 illustrate the braid fans F n and their shards Σ n when n = 3 and n = 4 respectively. As the 3-dimensional fan F 4 is difficult to visualize (as in Figure 1 (middle)), we use another classical representation in Figure 4 (left): we intersect F 4 with a unit sphere and we stereographically project the resulting arrangement of great circles from the pole 4321 to the plane. Each circle then corresponds to a hyperplane x i = x j with i < j, separating a disk where x i < x j from an unbounded region where x i > x j . In both  middle picture shows the shards Σ n (labeled by arcs), and the right picture represents the quotient fan F sylv by the sylvester congruence.
It turns out that the shards are precisely the pieces of the hyperplanes of H n needed to delimit the cones of the quotient fan F ≡ .
Theorem 9 ([Rea16a, Sect. 10.5]). For any lattice congruence ≡ of the weak order on S n , there is a subset of shards Σ ≡ ⊆ Σ n such that the interiors of the chambers of the fan F ≡ are precisely the connected components of H Σ ≡ .
Finally, we can describe the set of lattice congruences of the weak order on S n using the following poset on the shards Σ n . A shard Σ(i, j, S) is said to force a shard Σ(h, k, T ) if h i < j k and S = T ∩ ]i, j[. We denote this relation by Σ(i, j, S) Σ(h, k, T ). In terms of the corresponding arcs, the arc α of Σ(i, j, S) is a subarc of the arc β of Σ(h, k, T ), meaning that the endpoints of α are in between the endpoints of β, and the arc α agrees with the arc β between i and j. We call shard poset the poset (Σ n , ≺) of all shards ordered by forcing. The forcing relation and the shard poset on Σ 4 are illustrated on Figure 5. Example 11. For the sylvester congruence ≡ sylv , the corresponding shard ideal is the ideal Σ sylv = {Σ(i, j, ]i, j[) | 1 i < j n} of all up shards, i.e. those whose corresponding arcs pass above all dots in between their endpoints. Figures 3 and 4 (right) represent the quotient fans F ≡ sylv corresponding to the sylvester congruences ≡ sylv on S 3 and S 4 respectively. It is obtained • either by gluing the chambers C(σ) of the permutations σ in the same sylvester class, • or by cutting the space with the shards of To conclude these recollections on lattice congruences of the weak order, let us recall that the quotient fan F ≡ is essential if and only if the identity permutation is alone in its ≡-congruence class, or equivalently if Σ ≡ contains all basic shards Σ(i, i + 1, ∅) for i ∈ [n − 1] (this follows e.g. from [Rea04, Thm. 6.9]). We say that such a congruence is essential. If Σ ≡ does not contain the shard Σ(i, i + 1, ∅), then the quotient S n /≡ is isomorphic to the Cartesian product of the quotients S i /≡ and S n−i /≡ where ≡ and ≡ are the restrictions of ≡ to [1, i] and [i + 1, n] respectively. Any lattice congruence can thus be understood from its essential restrictions and we therefore focus on essential congruences.

Deformed permutahedra and removahedra
This section discusses the type cone of the braid fan F n . As they belong to the deformation cone of the permutahedron, we call the resulting polytopes deformed permutahedra. We also discuss a subfamily of specific deformed permutahedra called removahedra as they are obtained by deleting facets (instead of moving them).

Linear dependences in the braid fan
We start with classical considerations on the geometry of the braid fan. Remember that we have chosen a representative vector r(I) : = |I|1 − n1 I for the ray corresponding to each proper subset ∅ = I [n] (where 1 : = i∈[n] e i and 1 I : = i∈I e i ). We also set r(∅) = r([n]) = 0 by convention. Lemma 13. Let σ, τ be two adjacent permutations. Let ∅ = I [n] (resp. ∅ = J [n]) be the two proper subsets such that r(I) (resp. r(J)) is the ray of C(σ) not in C(τ ) (resp. of C(τ ) not in C(σ)). Then the linear dependence among the rays of the cones C(σ) and C(τ ) is r(I) + r(J) = r(I ∩ J) + r(I ∪ J).

Deformed permutahedra
We now consider the type cone of the braid fan, or in other words the deformation cone of the permutahedron, studied in details in [Pos09,PRW08]. The following classical statement is a consequence of Proposition 2 and Lemma 13. We naturally identify a vector h with coordinates indexed by the rays of the braid fan F n with a height function For instance, the height function of the permutahedron Perm n is given by A deformed permutahedron is a polytope whose normal fan coarsens the braid fan F n . It can be written as for some submodular function h : 2 [n] → R. We prefer the term deformed permutahedron to the term generalized permutahedron used by A. Postnikov in [Pos09,PRW08] (in particular because it also generalizes to other Coxeter groups [ACEP20]). Observe in particular that any quotientope P ≡ of Theorem 7 is a deformed permutahedron since the quotient fan F ≡ coarsens the braid fan F n by definition.

Removahedra
A removahedron is a deformed permutahedron obtained by deleting inequalities in the facet description of the permutahedron Perm n . In other words, it can be written as . Examples of removahedra include the permutahedron Perm n itself (remove no inequality), the associahedron Asso n (remove the inequalities that do not correspond to intervals), the graph associahedron Asso G [CD06,Dev09] if and only if the graph G is chordful [Pil17], and the permutreehedra [PP18] described below.
We say that a fan G with rays {r(I) | I ∈ I} is removahedral if G is the normal fan of the removahedron Remo I . We say that a lattice congruence ≡ of the weak order is removahedral if its quotient fan F ≡ is. The following example shows that some lattice congruence are not removahedral.
Since the only removed shard contains no ray in its interior, the rays of the quotient fan F ≡ are all rays of the braid fan F ≡ , so that the corresponding removahedron is the permutahedron Perm 4 which does not realize F ≡ .

Permutrees
We now recall the permutrees of [PP18] that generalize the binary trees and will be especially important in this paper. The presentation and pictures are borrowed from [PP18].

Combinatorics of permutrees
In an oriented tree T , we call parents (resp. children) of a node j the outgoing (resp. incoming) neighbors of j, and ancestor (resp. descendant) subtrees of j the connected components of the parents (resp. children) of j in T {j}. A permutree is an oriented tree T with nodes [n], such that • any node has either one or two parents and either one or two children, • if node j has two parents (resp. children), then all nodes in its left ancestor (resp. descendant) subtree are smaller than j while all nodes in its right ancestor (resp. descendant) subtree are larger than j. Figure 6 provides four examples of permutrees. We use the following conventions: • All edges are oriented bottom-up and the nodes appear from left to right in natural order.
• We decorate the nodes with , , , depending on their number of parents and children. The sequence of these symbols, from left to right, is the decoration δ of T . We let δ − : = {j ∈ [n] | δ j = or } and δ + : = {j ∈ [n] | δ j = or }.
• We draw a vertical red wall below (resp. above) the nodes of δ − (resp. of δ + ) to mark the separation between the left and right descendant (resp. ancestor) subtrees.
As illustrated in Figure 6, δ-permutrees extend and interpolate between various combinatorial families, including permutations when δ = n , binary trees when δ = n , and binary sequences when δ = n . In fact, permutrees arose by pushing further the combinatorics of Cambrian trees developed in [CP17] to provide combinatorial models to the type A Cambrian lattices [Rea06].
An edge cut in a permutree T is the partition (I J) of the nodes of T into the set I of nodes in the source set and the set J = [n] I of nodes in the target set of an oriented edge of T . Edge cuts play an important role in the geometry of the permutree fan defined below. As with the classical rotation operation on binary trees, there is a local operation on δpermutrees which only exchanges the orientation of an edge and rearranges the endpoints of two other edges. Namely, consider an edge i → j in a δ-permutree T . Let D denote the only (resp. the right) descendant subtree of node i if i / ∈ δ − (resp. if i ∈ δ − ) and let U denote the only (resp. the left) ancestor subtree of node j if j / ∈ δ + (resp. if j ∈ δ + ). Let S be the oriented tree obtained from T by just reversing the orientation of i → j and attaching the subtree U to i and the subtree D to j. The transformation from T to S is the rotation of the edge i → j.
Proposition 16 ( [PP18]). The rotation of an edge i → j in a δ-permutree T produces a δ-permutree S. Moreover, S is the unique δ-permutree with the same edge cuts as T , except the cut defined by i → j.
Define the increasing rotation graph as the directed graph whose vertices are the δpermutrees and whose arcs are increasing rotations T → S, i.e. where the edge i → j in T is reversed to the edge i ← j in S for i < j. See Figure 7.
The δ-permutree lattice specializes to the weak order when δ = n , the Tamari lattice when δ = n , the Cambrian lattices [Rea06] when δ ∈ { , } n and the boolean lattice when δ = n .
In fact, all permutree lattices are lattice quotients of the weak order, exactly as the Tamari lattice is the quotient of the weak order by the sylvester congruence. The δ-permutree congruence ≡ δ is the equivalence relation on S n defined equivalently as: • the relation whose equivalence classes are given by the sets of linear extensions of the δ-permutrees, • the relation whose equivalence classes are the fibers of the δ-permutree insertion, similar to the binary tree insertion and presented in detail in [PP18], • the transitive closure of the rewriting rules U ikV jW ≡ δ U kiV jW when j ∈ δ − and U jV ikW ≡ δ U jV kiW when j ∈ δ + , for some letters i < j < k and words U, V, W on [n], • the congruence of the weak order with arc ideal In other words, the corresponding arcs do not pass below any k ∈ δ − nor above any k ∈ δ + .
). The δ-permutree congruence ≡ δ is a lattice congruence of the weak order and the quotient S n /≡ δ is (isomorphic to) the δ-permutree lattice.
Note that the description of the shard ideal Σ δ above gives the following characterization of permutree congruences.  We conclude these recollections on combinatorics of permutrees with a natural order on all permutree congruences. For two decorations δ, δ ∈ { , , , } n , we say that δ refines δ and we write δ δ if δ i δ i for all i ∈ [n] for the order { , } . In this case, the δ-permutree congruence refines the δ -permutree congruence: σ ≡ δ τ implies σ ≡ δ τ for any two permutations σ, τ ∈ S n .

Geometry of permutrees
We finally recall the geometric constructions of [PP18] realizing the δ-permutree lattice.
The δ-permutree fan F δ is the quotient fan of the δ-permutree congruence. It has Two examples of δ-permutree fans are represented in Figure 9. The δ-permutree fan F δ specializes to the braid fan when δ = n , the (type A) Cambrian fans of N. Reading and D. Speyer [RS09] when δ ∈ { , } n , the fan defined by the hyperplane arrangement x i = x i+1 for each i ∈ [n − 1] when δ = n , and the fan defined by the hyperplane arrangement x i = x j for each 1 i < j n such that δ k = for all i < k < j when δ ∈ { , } n . Decoration refinements translate to permutree fan refinements: if δ δ , then the δ-permutree fan F δ refines the δ -permutree fan F δ .
The δ-permutreehedron PT δ is the polytope defined equivalently as: • the convex hull of the points j∈ for all subsets I in I δ .
Note that decoration refinements translate to permutreehedra inclusions: if δ δ , then the permutreehedron PT δ is obtained by deleting inequalities in the facet description of the permutreehedron PT δ , so that PT δ ⊆ PT δ as illustrated in Figure 11. In particular, Figure 11: The δ-permutreehedra, for all decorations δ ∈ · { , , , } 2 · . [PP18, Fig. 16] all facet-defining inequalities of the permutreehedron PT δ are facet-defining inequalities of the permutahedron Perm n , which can be rephrased as follows.

Removahedral congruences
The main goal of this section is to show Theorem 1. An important step of the proof is the description of the rays of the quotient fans, provided in Section 3.1. While elementary, we are not aware that a characterization of these rays appeared in the literature. We then proceed to prove Theorem 1 (i) in Section 3.2 and Theorem 1 (ii) in Section 3.3.

Rays of the quotient fan
Fix a ray r(I) of the braid fan F n corresponding to a proper subset ∅ = I [n]. The shards of Σ n containing r(I) were characterized by a simple combinatorial criterion in [PS19, Lem. 5]. Here, we need to characterize which shards of Σ n contain r(I) in their relative interior. We associate with I the set Σ I of n − 2 shards containing the |I| − 1 down shards joining two consecutive elements of I and the n − |I| − 1 up shards joining two consecutive elements of [n] I, that is Lemma 22. The n − 2 shards of Σ I are the only shards containing r(I) in their relative interior.
Proof. Recall that • the ray r(I) is given by |I|1 − n1 I , thus x i < x j for i ∈ I and j / ∈ I, • the relative interior of Σ(i, j, S) is given by Proof. If Σ ≡ contains Σ I , then the quotient fan F ≡ contains n − 2 shards which intersect along r(I), so that r(I) is a ray of F ≡ . The converse can be derived from [Rea05, Prop. 5.10] or [Rea11, Prop. 7.7]. Let us just provide a sketchy argument. Let S I be the interval of permutations σ of [n] such that cone C(σ) contains r(I) (or equivalently σ([|I|]) = I). Let ≡ I denote the subcongruence of ≡ induced by S I . The basic shards of ≡ I are the shards of Σ I . Since Σ ≡ does not contain Σ I , the congruence ≡ I is not essential, so that r(I) is not a ray of F ≡ .

Removahedral congruences
We are now ready to prove the following statement, which is a more precise reformulation of Theorem 1 (i).
Theorem 24. The only essential removahedral congruences are the permutree congruences.
Note that we focus here on essential congruences. However, as already mentioned, nonessential congruences can be understood from their essential restrictions. For arbitrary congruences, Theorem 24 says that the essential restrictions of removahedral congruences are permutree congruences.
We learned from Corollary 21 that permutree congruences are essential removahedral congruences, and we want to prove the opposite direction. We thus assume by contradiction that there exists an essential removahedral congruence ≡ which is not a permutree congruence.
Lemma 25. The generating set Π ≡ of the lower ideal Σ n Σ ≡ contains at least one shard of the form (i, j, ∅) or (i, j, ]i, j[) for i j − 3.
Proof. Since ≡ is essential, Π ≡ contains no shard of length 1. Since ≡ is not a permutree congruence, Π ≡ must contain a shard of length distinct from 2 by Proposition 19. Decompose the set of shards Π ≡ into the subset Π =2 ≡ of shards of length 2 and the subset Π >2 ≡ of shards of length greater than 3. Let ≡ δ denote the permutree congruence defined by Π ≡ δ = Π =2 ≡ . If Π >2 ≡ contains only mixed shards, then ≡ and ≡ δ contain the same up and down shards, since mixed shards only force mixed shards. This implies by Lemma 23 that the quotient fans F ≡ and F δ have the same rays. The corresponding removahedron is thus the permutreehedron PT δ which does not realize the quotient fan F ≡ so that ≡ is not removahedral. The statement follows.
We assume now that there are i j − 3 such that the shard Σ(i, j, ]i, j[) is one of the generators of the lower ideal Σ n Σ ≡ . The proof for the shard Σ(i, j, ∅) is symmetric. We consider the following five subsets of [n]: Therefore, the lattice congruence ≡ is not removahedral.
We now come back to the general situation. We need the following three observations. the electronic journal of combinatorics 28(4) (2021), #P4.8 Lemma 27. The rays r(I), r(J), r(K), r(L), and r(M ) are all rays of the quotient fan F ≡ .
Proof. By Lemma 22, the only non-basic shards containing these rays in their interior are • Σ(i + 1, j, ]i + 1, j[) for I, Since all these shards are in Σ ≡ by minimality of Σ(i, j, ]i, j[), the result follows by Lemma 23. For completeness, let us provide a less intuitive but more formal alternative argument for this statement. Consider a sequence of permutations starting with the permutation σ : = [i + 2, . . . , j − 2, j − 1, i + 1, i, . . . , 1, j, . . . , n] and ending with the permutation τ : = [i + 2, . . . . . . , j − 2, j − 1, i + 1, j, . . . , n, i, . . . , 1], and obtained by transposing at each step two values k i and j at consecutive positions. In other words, all the permutations in the sequence start by [i + 2, . . . , j − 2, j − 1, i + 1] and end with a shuffle of [i, . . . , 1] with [j, . . . , n]. At each step, the interval ]k, [ between the two transposed values always appears before the position of the transposition. Therefore, the chambers corresponding to the two permutations before and after the transposition are separated by the shard Σ(k, , ]k, [), which does not belong to Σ ≡ since it is forced by Σ(i, j, ]i, j[). It follows that the cones of all these permutations, and in particular those of σ and τ , belong to the same chamber C of the quotient fan F ≡ . This chamber C contains the rays r(I), r(K), r(L) and r(M ) since the subsets I and M (resp. K, resp. L) are initial intervals of all permutations in the sequence (resp. of σ, resp. of τ ). We prove similarly that r(J), r(K), r(L) and r(M ) belong to a chamber D by considering a sequence of permutations starting with [i + 2, . . . , j − 2, i + 1, j − 1, i, . . . , 1, j, . . . , n] and ending with [i + 2, . . . , j − 2, i + 1, j − 1, i, . . . , 1, j, . . . , n].
Lemma 29. Let h • (I) = n|I|(n − |I|)/2 be the height function of the classical permutahedron Perm n . Then • r(I) + r(J) = r(K) + r(L) + r(M ), Proof. Immediate computations from the cardinalities Observe that the inequality h

Permutree congruences are strongly removahedral
We now prove that the permutree congruences are removahedral in a stronger sense. Namely, we show that we obtain polytopes realizing the permutree fans by deleting inequalities in the facet description of any polytope realizing the braid fan, not only the classical permutahedron Perm n , as stated in Theorem 1 (ii). We start by a structural observation on the exchanges in permutree fans. Proof. Consider the two δ-permutrees T and S whose chambers are C(T ) = R 0 R and C(S) = R 0 S. Let i → j denote the edge of T that is rotated to the edge j → i in S. Up to swapping the roles of I and J, we can assume that i < j. We denote by U , D, L, L, R, R the subtrees of T and S as illustrated on the figure on the right. Note that some of the subtrees L, L, R, R might not exist if δ i = or δ j = . We then just assume that they are empty. Since the rays of the cone C(T ) are given by |I|1 J − |J|1 I for all edge cuts of (I J) of T , and the unique edge cut that differs from T to S is that corresponding to the edge ij by Proposition 16, we obtain that and therefore that where I δ = {∅ = I [n] | r(I) is a ray of F δ } (see Proposition 32 for a characterization). In other words, we obtain a polytope realizing the δ-permutree fan F δ by deleting inequalities in the facet description of any polytope realizing the braid arrangement F n .

Type cones of permutree fans
In this section, we provide a complete facet description of the type cone TC(F δ ) of the δ-permutree fan F δ . We first describe the rays of F δ in Section 4.1, then the pairs of exchangeable rays of F δ in Section 4.2, and finally the facets of the type cone TC(F δ ) in Section 4.3. We conclude by a description of kinematic permutreehedra for the permutree fans F δ whose type cone is simplicial in Section 4.4.

Rays of permutree fans
Specializing Lemma 23 to the δ-permutree congruence ≡ δ , we obtain the following description of the rays of the δ-permutree fan F δ announced in Section 2.4.2.
Proposition 32. A ray r(I) is a ray in the δ-permutree fan F δ if and only if for all a < b < c, if a, c ∈ I then b / ∈ δ − I, and if a, c / ∈ I then b / ∈ δ + ∩ I.
Example 34. Specializing Proposition 32, we recover the following classical descriptions: • when δ = n , the rays of the braid fan F n are all proper subsets ∅ = I [n], • when δ = n , the rays of F n are all proper intervals [i, j] of [n], (equivalently, one can think of the interval [i, j] as corresponding to the internal diagonal (i − 1, j + 1) of a polygon with vertices labeled 0, . . . , n + 1), • when δ = n , the rays of An alternative argument would be to observe directly that these conditions are necessary and sufficient to allow the construction of a δ-permutree with an edge whose cut is (I [n] I).
Example 37. Specializing the formula of Corollary 35, we recover the following classical numbers: • when δ = n , the braid fan F n has 2 n − 2 rays, • when δ = n , the fan F n has n+1 2 − 1 rays (equalling the number of internal diagonals of the (n + 2)-gon), • when δ = n , the fan F n has 2n − 2 rays.
Proof of Corollary 35. To choose a ray r(I) of F δ , we proceed as follows: • We choose the last position i (resp. first position j) such that 1, . . . , i (resp. j, . . . , n) all belong to I or all belong to [n] I. Note that 1 i < j n.
• For any i < k < j, since |{i, i+1}∩I| = 1 and |{j−1, j}∩I| = 1, the characterization of Proposition 32 imposes that k ∈ I if k ∈ δ − , and k / ∈ I if k ∈ δ + . This is impossible if δ k = (explaining the condition over the sum), and leaves two choices if δ k = (explaining the power of 2).
• If i + 1 < j, then the presence of i + 1 (resp. j − 1) in I requires the absence of i (resp. j) in I and vice versa, so there is no choice left.
• If i + 1 = j, we have counted only one ray while both subsets [1, i] and [j, n] indeed correspond to rays of F δ .

Exchangeable rays of permutree fans
An immediate corollary of Proposition 30 is that the linear dependence between the rays of two adjacent chambers C : = R 0 R and D : = R 0 S of F δ with R S = {r} and S R = {s} only depend on the rays r and s, not on the chambers C and D. This property is called unique exchange relation property in [PPPP19] and allows to describe the type cone by inequalities associated with exchangeable rays rather than with walls. We therefore proceed in identifying the pairs of exchangeable rays of F δ . We consider two subsets I, J ∈ I δ , i.e. proper subsets ∅ = I, J [n] such that r(I) and r(J) are rays of the δ-permutree fan F δ , as characterized in Proposition 32. • when δ = n , we have I = [h, j[ and J = ]i, k] for some 1 h i < j k n, (equivalently, the internal diagonals (h − 1, j) and (i, k + 1) of the (n + 2)-gon intersect), • when δ = n , we have I = [1, i] and J = ]i, n] for some 1 i < n.
the electronic journal of combinatorics 28(4) (2021), #P4.8 Proof of Proposition 39. We first prove that the conditions of Proposition 39 are necessary for the rays r(I) and r(J) to be exchangeable in the δ-permutree fan F δ . We keep the notations of the proof of Proposition 30. Remember that we had I = {i} ∪ D ∪ L ∪ L and J = {j} ∪ D ∪ R ∪ R. Since L ∪ L < i < j < R ∪ L, we obtain that i = max(I J) and j = min(J I) indeed satisfy (i). Moreover, I J = {i} ∪ L ∪ L is restricted to {i} if δ i = (because the subtrees L and L must then be empty), and similarly J I = {j} ∪ R ∪ R is restricted to {j} if δ j = , wich shows (ii). Finally, if there is i < k < j such that k ∈ δ − (I ∩ J) (resp. k ∈ δ + ∩ (I ∩ J)), then the edge ij crosses the red wall below k (resp. above k), which shows (iii).
Assume now that I and J satisfy the conditions of Propositions 32 and 39. We construct two δ-permutrees T and S, connected by the rotation of the edge ij whose edge cut in T is I and in S is J. For this, we first pick an arbitrary permutree, that we denote by D (resp. U , resp. L, resp. R), for the restriction of the decoration δ to the subset I ∩ J (resp. [n] (I ∪ J), resp. I J {i}, resp. J I {j}). We then construct an oriented tree T on [n] starting with an edge i → j and placing • D as the only (resp. the right) descendant subtree of • U as the only (resp. the left) ancestor subtree of j if j / ∈ δ + (resp. if j ∈ δ + ), • L as the left descendant (resp. ancestor) subtree of • R as the right descendant (resp. ancestor) subtree of j if j ∈ δ − (resp. if j / ∈ δ − ).
Note that there is only one way to place these subtrees. For instance, to place D, we connect the leftmost upper blossom of D to the only (resp. the right) lower blossom of i if i / ∈ δ − (resp. if i ∈ δ − ). We claim that the conditions of Propositions 32 and 39 ensure that T is a δ-permutree. Observe first that it indeed forms a tree since the permutree L (resp. R) is empty if δ i = (resp. δ j = ) by Proposition 39 (ii). Hence, we just need to show that no edge of T crosses a red wall below a node i ∈ δ − or above a node i ∈ δ + . Since all nodes of L (resp. R) are smaller than i (resp. j) by Proposition 39 (i), and there is no red wall above (resp. below) the nodes of D (resp. U ) between i and j by Proposition 39 (iii), the edge i → j crosses no red wall. Consider now an edgeof L ∪ {i} with < . It cannot cross a red wall emanating from a node r of R ∪ {j} since i < j r, nor from a node u of U since otherwise we would have < u < with , ∈ I and u ∈ δ − I contradicting Proposition 32, nor from a node d of D since otherwise we would have < d < with , / ∈ J and d ∈ δ + ∩ J contradicting Proposition 32. We prove similarly that no edge in D ∪ {i}, nor in U ∪ {j}, nor in R ∪ {j} crosses a red wall. This closes the proof that T is a δ-permutree.
Finally, denote by S the δ-permutree obtained by the rotation of the edge i → j in T . Observe that the construction is done so that the edge i → j in T has cut (I [n] I) while the edge j → i in S has cut (J [n] J). It follows that the rays r(I) and r(J) are exchangeable in the adjacent chambers C(T ) and C(S) of the δ-permutree fan F δ .
Example 44. Specializing the formula of Corollary 42, we recover the following classical numbers: • when δ = n , the braid fan F n has 2 n−2 n 2 pairs of exchangeable rays, • when δ = n , the fan F n has n+2 4 pairs of exchangeable rays (equalling the number of quadruples of vertices of the (n + 2)-gon), • when δ = n , the fan F n has n − 1 pairs of exchangeable rays.
Proof of Corollary 42. A pair of exchangeable rays in the δ-permutree fan F δ is a pair of proper subsets ∅ = I, J [n] satisfying the conditions of Propositions 32 and 39. We choose such a pair of subsets as follows: • We first choose 1 i < j n and will have i = max(I J) and j = min(J I) (to fulfill Proposition 39 (i)).
• For all i < k < j, we must have k ∈ I ∩ J if k ∈ δ − and k ∈ [n] (I ∪ J) if k ∈ δ + (to fulfill Proposition 39 (iii)). This is impossible if δ k = (explaining the condition over the sum), and leaves two choices if δ k = (explaining the power of 2).
• For all k < i, we must have k ∈ I if δ i ∈ δ + , and k / ∈ J if δ i ∈ δ − (to fulfill Proposition 32). Moreover, k ∈ I J implies δ i = (to fulfill Proposition 39 (ii)). Thus, k must lie in Moreover, the choices in the last three cases are handled by the function Ω.
• For j < k, the argument is symmetric.

Facets of types cones of permutree fans
In view of the unique exchange property of the δ-permutree fan F δ , each pair of exchangeable rays of F δ yields a wall-crossing inequality for the type cone TC(F δ ). However, not all pairs of exchangeable rays yield facet-defining inequalities of TC(F δ ). The characterization of the facets of TC(F δ ) is very similar to that of the exchangeable rays, only point (ii) slightly differs. Example 47. Specializing Proposition 45, we recover that all pairs of exchangeable rays of F δ described in Example 41 define facets of the type cone TC(F δ ) when δ = n or δ = n . In contrast, when δ = n , only the pairs of intervals {[i, j[, ]i, j]} for some 1 i < j n correspond to facets of the type cone TC(F n ) (equivalently, the internal diagonals (i − 1, j) and (i, j + 1) of the (n + 2)-gon that just differ by a shift). Proof of Proposition 45. We consider two exchangeable rays r(I) and r(J), so that I and J satisfy the conditions of Propositions 32 and 39. We will show that they satisfy the additional condition of Proposition 45 (ii) if and only if the wall-crossing inequality corresponding to the exchange of r(I) and r(J) defines a facet of the type cone TC(F δ ).
Let us detail the case |I J| > 1 and δ i = , the other situations being symmetric. As usual, we define For each of the subsets D, U , L and R, we choose an arbitrary permutree, and we construct δ-permutrees T and S as in the proof of Proposition 39 and as illustrated in Figure 12, so that rotation from T to S exchanges r(I) to r(J). Note that we voluntarily placed R at the same level as node j, as it can be the right ancestor or descendant subtree of j, depending on δ j . We know that the wall-crossing inequality corresponding to this rotation is Consider now the δ-permutrees V and W of Figure 12. In the tree V (resp. W ), the leftmost lower blossom of L is connected to j (resp. i), and the rightmost upper blossom of L is connected to the blossom of U to which j was connected in T . Note that these are indeed δ-permutrees since δ i = . The rotation between V and W yields the wall-crossing inequality where I : = {i} ∪ D = {i} ∪ (I ∩ J). Note that we could also have checked that I and J satisfy the conditions of Propositions 32 and 39. Consider now the δ-permutrees X and Y of Figure 12. We have already seen that Y is indeed a δ-permutree because it is just equal to V . The same arguments show that X is also a δ-permutree. Consider now the rotation of the edge joining L to j. If i and j are connected to the same node in L, then this rotation gives the δ-permutree Y . Otherwise, this rotation moves a part of L in between j and U and leaves the remaining part of L in between i and j. Rotating again and again the edge between this remaining part of L and j, we finally obtain the δ-permutree Y . More formally, we can consider the sequence of rotations between the δ-permutrees X k and Y k illustrated in Figure 12, where at each step we use X k+1 = Y k . In these δ-permutrees, we have L k L k = L. Moreover, L k+1 is obtained from L k by deleting the node connected to j in X k and all its left ancestors and descendants. Hence, starting with X 0 = X, we will end with Y p = Y . Summing the wall-crossing inequalities corresponding to the rotations between X k and Y k , we thus obtain the inequality where J : = {i} ∪ J. Finally, observe that I = I ∩ J and J = I ∪ J. We therefore obtain that the wallcrossing inequality (1) can be expressed as the sum of inequalities (2) and (3), showing that I and J do not define a facet of the type cone TC(F δ ).
Conversely, assume that I and J satisfy the conditions of Proposition 45. To prove that the wall-crossing inequality associated with {I, J} indeed defines a facet of the type cone TC(F δ ), we exhibit a point p that satisfies the wall-crossing inequality associated and we consider the point p ∈ R I defined by where λ and µ are arbitrary scalars such that λ > | z | n(K, L) | for any pair {K, L} of subsets and 0 < µ |∇(I, J)| < z | z . We will prove that the point p satisfies the desired inequalities.
For this, consider a pair {K, L} that fulfills the conditions of Proposition 45. Observe that  Finally, we conclude by proving our claim that if ∇(K, L) ∇(I, J), then the sets {I, J, I ∪ J, I ∩ J} and {K, L, K ∪ L, K ∩ L} are disjoint. Up to reversing the roles of I and J (resp. K and L), we assume that i : = max(I J) < min(J I) = : j (resp. k : = max(K L) < min(L K) = : ). As already observed earlier, the inclusion ∇(K, L) ∇(I, J) implies the inclusions K L ⊆ I J and L K ⊆ J I, so that k i < j . It turns out that the conditions of Proposition 45 actually imply that K L = I J and L K = J I, so that k = i and j = . Indeed, • similarly, if J = L, then I = K.
In these four cases, we obtained that I = K and J = L, which contradicts our assumption that ∇(K, L) = ∇(I, J). Finally, observe that The proof of the following consequence of Proposition 45 is almost identical to that of Corollary 42.
Example 51. Specializing the formula of Corollary 49, we recover the following classical numbers: • when δ = n , the type cone TC(F n ) has 2 n−2 n 2 facets, • when δ = n , the type cone TC(F n ) has n 2 facets (equalling the number of squares of the form (i − 1, i, j, j + 1) in the (n + 2)-gon), • when δ = n , the type cone TC(F n ) has n − 1 facets.
Corollary 53. The type cone TC(F δ ) is simplicial if and only if δ k = for any k ∈ ]1, n[.
2. the facets of its type cone TC(F δ ) by F .