Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.


INTRODUCTION
Given a permutation w of [n] def = {1, 2, . . . , n}, an inversion of w is a pair of indices for which the corresponding values of w are out of order. In other words, the number of inversions of w is a measure of disorder introduced by w. A permutation is characterized by its inversion set, i.e. the set of pairs encoding the locations of the inversions.
Containment of inversion sets introduces a partial order on the set S n of all permutations of [n]; the weak order. This partial order has many remarkable properties. For instance, it is a lattice [23,46]. The diagram of the weak order is the graph on the vertex set S n in which two permutations are related by an edge if they differ by swapping a descent, i.e. an inversion whose corresponding values are adjacent integers. By construction, this diagram is isomorphic to the 1-skeleton of the permutohedron [19].
Perhaps an even more remarkable property of the weak order on S n is the fact that it is a semidistributive lattice [16]. This means that every permutation has a canonical representation as a join of permutations; thus effectively solving the word problem for these lattices. The members of these canonical join representations are join-irreducible permutations, i.e. permutations with a unique descent.
In [12], a property stronger than semidistributivity was established for the weak order on S n . It was shown that weak order lattices are congruence uniform, which ensures a bijective connection between join-irreducible permutations and joinirreducible lattice congruences, i.e. certain equivalence relations on S n compatible with the lattice structure.
N. Reading gave a combinatorial description of the canonical join representations in the weak order in terms of noncrossing arc diagrams [37]. Each joinirreducible permutation of S n corresponds to a unique arc connecting two distinct elements of [n], and a certain forcing order on these arcs can be used to characterize quotient lattices of the weak order.
One of these quotient lattices is the Tamari lattice, first introduced in [42] via a rotation transformation on binary trees. When considered as a quotient lattice of the weak order on S n , the Tamari lattice-denoted by Tam(n)-arises as the subposet induced by 231-avoiding permutations, i.e. permutations whose one-line notation does not contain a subword that normalizes to 231 [10,34].
The Tamari lattices have an even richer structure than the weak order on permutations [31]. The Tamari lattices inherit semidistributivity and congruenceuniformity from the weak order, but they are also trim, i.e. extremal and left modular [11,26]. The first property implies that their number of join-irreducible elements is as small as possible, and the second property entails some desirable topological properties.
The noncrossing arc diagrams representing the elements of Tam(n) are precisely the noncrossing partitions of [n] introduced in [24]; see [37]. Then, generalizing a geometric construction by N. Reading, there is a natural way to reorder the elements of Tam(n) which turns out to agree with the refinement order on noncrossing partitions [3,36].
Let us expand on this construction a little bit. Since the weak order is congruence uniform, we may use a perspectivity relation to label the edges in its diagram by join-irreducible permutations. With any permutation w, we can associate a particular interval in the weak order by taking the meet of the elements covered by w. The core label set of w is the set of labels appearing in this interval and the core label order orders S n with respect to containment of these core label sets.
Note that this construction is purely lattice-theoretic and depends only on a (finite) lattice L and a labeling of the diagram of L. Under certain hypotheses on this labeling, we can associate a core label order CLO(L) with any labeled lattice. A study of this core label order for congruence-uniform lattices was carried out in [29].
The parabolic Tamari lattice is the restriction of the weak order to the subset of S α consisting of those permutations avoiding certain 231-patterns. By [30,Theorem 1], the resulting partially ordered set, denoted by Tam(α), is a quotient lattice of the weak order on S α . We set out for a structural study of these lattices and certain related structures. Our first main result establishes that Tam(α) is congruence uniform and trim. Theorem 1.1. For all n > 0 and every composition α of n, the lattice Tam(α) is congruence uniform and trim.
Since Tam(α) is congruence uniform, we may consider its core label order. Using a modification of Reading's noncrossing arc diagrams, we may relate the core label order of Tam(α) to the refinement order on certain set partitions of [n]. Exploiting the fact that Tam(α) is a quotient lattice of the weak order allows us to prove the following structural property of CLO Tam(α) . Theorem 1.2. Let n > 0 and let α be a composition of n. The core label order of Tam(α) is a meet-semilattice. It is a lattice if and only if α = (n) or α = (1, 1, . . . , 1).
As a consequence of Theorem 1.1, Tam(α) is extremal and thus admits a canonical representation as a lattice of set pairs defined on a certain directed graph; the Galois graph [26,45]. We give a combinatorial characterization of this graph in terms of join-irreducible permutations. We denote the (unique) join-irreducible permutation of Tam(α), whose only descent is (a, b), by w a,b . Theorem 1.3. Let n > 0 and let α be a composition of n. The Galois graph of Tam(α) is isomorphic to the directed graph whose vertices are the join-irreducible elements of Tam(α) and in which there exists a directed edge w a,b → w a ,b if and only if w a,b = w a ,b and • either a and a belong to the same α-region and a ≤ a < b ≤ b, • or a and a belong to different α-regions and a < a < b ≤ b, where a and b belong to different α-regions, too.
This article is organized as follows. In Section 2, we define the main objects considered here: (parabolic quotients of) the symmetric group, the weak order and the (parabolic) Tamari lattices. In order to keep the combinatorial flow of this article going, we have collected the necessary order-and lattice-theoretic concepts in Appendix A. We recommend to read the combinatorial parts of this article in order and refer to the appendix whenever unknown terminology is encountered.
In Section 3, we prove that the parabolic Tamari lattices are congruence uniform and study their associated core label order. We investigate the join-irreducible elements in the parabolic Tamari lattices in Section 4 and prove our main results. We conclude this article with an enumerative observation relating the generating function of the Möbius function in the core label order of the parabolic Tamari lattices and the generating function of antichains in certain partially ordered sets in Section 5.   w ∈ S n and i ∈ [n] we write w i instead of w(i). The one-line notation of w is the string w 1 w 2 . . . w n . For i, j ∈ [n] with i < j, the permutation that exchanges i and j and fixes everything else is a transposition; denoted by t i,j . If j = i + 1, then we write s i instead of t i,i+1 . The one-line notation of w • t i,j is the same as the one-line notation of w except that the i th and the j th entries are swapped. The one-line notation of t i,j • w is the same as the one-line notation of w except that the positions of the values i and j are swapped.
A (right) inversion of w is a pair (i, j) with i < j and w i > w j . A (right) descent of w is a pair (i, j) with i < j and w i = w j + 1. Let Inv(w) denote the set of (right) inversions of w, and let Des(w) denote the set of (right) descents of w.
Remark 2.1. We wish to emphasize that we consider "inversions" with respect to positions, meaning that composing on the right with a transposition swaps the entries in positions i and j.
It is much more common in Coxeter-Catalan theory to consider left inversions with respect to values. The reason for choosing this convention is the fact that it is more convenient for us to spot membership in parabolic quotients this way.
Ordering permutations of [n] with respect to containment of their (right) inversion sets yields the (left) weak order, denoted by ≤ L . For any subset X ⊆ S n we write Weak(X) def = (X, ≤ L ) for the set X partially ordered by ≤ L . Figure 1a shows Weak(S 4 ).
It follows from definition of the (left) weak order that two permutations u, v satisfy u L v if and only if Inv(v) \ Inv(u) = (i, j) and v i = v j + 1. In other words, u L v if and only if v = s u i • u and v has more inversions than u. Theorem 2.2 ([12, 23, 46]). For all n > 0, Weak(S n ) is a congruence-uniform lattice.
See Section A.2 for the definition of a congruence-uniform lattice. A consequence of Theorem 2.2 is the existence of a least element (the identity e def = 1 2 . . . n) and a greatest element (the long element w o def = n n−1 . . . 1) in Weak(S n ).

231-avoiding permutations and the Tamari lattice.
We now exhibit an important sub-and quotient lattice of Weak(S n ), the Tamari lattice Tam(n).
A 231-pattern in a permutation w ∈ S n is a triple (i, j, k) with i < j < k and w k < w i < w j . Then, w is 231-avoiding if it does not have a 231-pattern. Let S n (231) denote the set of 231-avoiding permutations of [n].
The Tamari lattice is, as far as we are concerned, the poset . This poset is named after D. Tamari, who introduced it in [42] via a partial order on binary trees and proved its lattice property. The fact that Weak S n (231) incarnates Tam(n) follows from [10,Theorem 9.6]. Figure 1b shows Tam(4).
The next theorem states some important, lattice-theoretic properties of Tam(n). See Sections A.2 and A.5 for the corresponding definitions. ([10, 11, 22, 26, 34]). For all n > 0, Tam(n) is a sublattice and a quotient lattice of Weak(S n ). Moreover, Tam(n) is trim and congruence uniform.

2.3.
Parabolic quotients of the symmetric group. Let α = (α 1 , α 2 , . . . , α r ) be a composition of n, and define p 0 . We define the parabolic subgroup of S n with respect to α by The symmetric group S n is generated by its set {s 1 , s 2 , . . . , s n−1 } of adjacent transpositions, and therefore any w ∈ S n can be written as a product of the s i 's. The length of w is the minimal number of adjacent transpositions needed to form w as such a product. It follows from [46, Proposition 2.1] that the length of w equals its number of inversions.
Let S α be the set of all minimal-length representatives of the left cosets of G α in S n , i.e.
The elements of S α are α-permutations. For i ∈ [r], the set p i−1 +1, p i−1 +2, . . . , p i is the i th α-region. We indicate α in the one-line notation of w ∈ S n either by coloring every α-region with a different color or by separating α-regions by a vertical bar. Figure 2a highlights the elements of S (1,2,1) in Weak(S 4 ).
The set S α behaves quite well with respect to left weak order.

Theorem 2.4 ([8]
). For all n > 0 and every composition α of n, Weak(S α ) is a principal order ideal in Weak(S n ). Consequently, Weak(S α ) is a congruence-uniform lattice.
A consequence of Theorem 2.4 is the existence of a greatest element in Weak(S α ). This element is denoted by w o;α and its one-line notation is of the following form: The vertical bars have no impact on the one-line notation, they shall only help separating the α-regions.   Clearly, if α = (1, 1, . . . , 1) is a composition of n, then S (1,1,...,1) = S n .

2.4.
Parabolic 231-avoiding permutations and the parabolic Tamari lattice. Generalizing the constructions from Section 2.2, we now identify a particular quotient lattice of Weak(S α ). An (α, 231)-pattern in an α-permutation w ∈ S α is a triple (i, j, k) with i < j < k all in different α-regions such that w k < w i < w j and w i = w k + 1. Then, w is (α, 231)-avoiding if it does not have an (α, 231)-pattern. Let S α (231) denote the set of (α, 231)-avoiding permutations of [n].
Remark 2.6. In general, we need to distinguish cover relations in Tam(α) from cover relations in Weak(S α ), and we do so by using α (resp. L ). The reason is that for u, v ∈ S α , u α v does not necessarily imply u L v; see Figure 2. More generally, we indicate poset-and lattice-theoretic notions in Tam(α) with a subscript "α", and in Weak(S α ) with a subscript "L".

THE α-TAMARI LATTICES ARE CONGRUENCE-UNIFORM
We start right away with the proof that Tam(α) is congruence uniform. Proof. By Theorem 2.2, Weak(S n ) is congruence uniform and by Theorems 2.4 and 2.5, Tam(α) is a quotient lattice of an interval of Weak(S n ). By [15,Theorem 4.3], congruence-uniformity is preserved under passing to sublattices and quotient lattices. This proves the claim. Proof. This follows from Proposition 3.1 and Theorem A.8.
3.1. Noncrossing α-partitions. Our next goal is a combinatorial description of the canonical join representations in Tam(α). In preparation, we introduce another combinatorial family parametrized by α.
An α-arc is a pair (a, b), where 1 ≤ a < b ≤ n and a, b belong to different αregions. Two α-arcs (a 1 , b 1 ) and (a 2 , b 2 ) are compatible if a 1 = a 2 or b 1 = b 2 , and the following is satisfied: then either a 1 and a 2 lie in the same α-region or b 1 and a 2 lie in the same α-region; (NC2): if a 1 < a 2 < b 2 < b 1 , then a 1 and a 2 lie in different α-regions. An α-partition is a set partition of [n], where no block intersects an α-region in more than one element. Let Π α denote the set of α-partitions of [n].
Let P ∈ Π α , and let B ∈ P be a block. If a, b ∈ B, then we write a ∼ P b. A bump of P is a pair (a, b) such that a, b ∈ B and there is no c ∈ B with a < c < b.
We graphically represent an α-arc (a, b) as follows. We draw n nodes on a horizontal line, label them by 1, 2, . . . , n from left to right, and group them together according to α-regions. Now we draw a curve leaving the node labeled a to the bottom, staying below the α-region containing a, moving up and over the subsequent α-regions until it enters the node labeled b from above.
An α-partition is noncrossing if and only if its bumps can be drawn in this manner such that no two curves intersect in their interior. Likewise, any collection of pairwise compatible α-arcs corresponds to a noncrossing α-partition whose blocks are given by the connected components of the graphical representation of the αarcs. Figure 3a shows a graphical representation of a noncrossing α-partition. . For all n > 0 and every composition α of n, the sets S α (231) and Nonc(α) are in bijection. This bijection sends descents to bumps.
Let Φ α denote the bijection from Theorem 3.4. If w ∈ S α (231), then Des(w) is a collection of pairwise compatible α-arcs; and thus corresponds to some The smaller parabolic noncrossing partitions P 1 ∈ Nonc (1, 2, 1, 2, 1) (top) and P 2 ∈ Nonc (2, 2, 1, 1) (bottom).  Nonc(α). Conversely, if P ∈ Nonc(α), then we define an acyclic binary relation R P on the blocks of P by setting (B, B ) ∈ R P if and only if there exists an α-arc (a, b) such that a, b ∈ B and a < min B < b for a and min B in different α-regions.
Let O P denote the reflexive and transitive closure of R P . Without loss of generality, we may assume that , and w 1 = |X|, where X is the union of the blocks in the order filter of O P generated by B 1 . The remaining values for w are determined by considering two smaller parabolic noncrossing partitions P 1 and P 2 , where P 1 is the restriction of P to X \ B 1 , and where P 2 is the restriction of P to [n] \ X. Note that O P 1 and O P 2 are induced subposets of O P . See Figure 3 for an illustration. Tam(α). We now explain how to use noncrossing α-partitions to describe canonical join representations in Tam(α). Essentially, we are going to prove that, for w ∈ S α (231), the set of bumps of Φ α (w) determines the canonical join representation of w in Tam(α).

Proposition 3.5.
For all n > 0 and every composition α of n, the canonical join repre- We now gather some ingredients required for the proof of Proposition 3.5.
By definition of the weak order, n L = Des(w) . Since Weak(S α ) is congruence uniform, n L equals the number of canonical joinands of w in Weak(S α ) by Corollary A.11. If w ∈ S α (231), then π ↓ α (w) = w and Proposition A.12 implies that n L is the number of canonical joinands of w in Tam(α), which is also n α . Proof. An element w ∈ Tam(α) is join irreducible if and only if it covers a unique element. By Lemma 3.6, w is join irreducible if and only if it has a unique descent. By Theorem 3.4, Φ α (w) is a noncrossing α-partition with a unique bump, and thus corresponds to an α-arc. Since Φ α is a bijection, the claim follows. Corollary 3.8. Let (a, b) be an α-arc, where a belongs to the j th α-region. The corresponding join-irreducible element of Tam(α) is w a,b ∈ S α (231) given by

The inversion set of w a,b is
. Corollary 3.8 implies that the inversion set of w a,b can be read off easily from Φ(w a,b ). In fact, the first components of an inversion of w a,b are the nodes that lie weakly to the right of a and weakly above the arc connecting nodes a and b, the second components are the nodes that lie weakly below this arc. This is illustrated in the following example in the case n = 8, a = 2, b = 6; see also Figure 4.
in whose one-line notation the entries in positions a and b are swapped-satisfies u 1 L v, and it follows (a, b) / ∈ Inv(u 1 ).
Recall the definition of perspective cover relations from Section A.2.
. Then, u α v and w a,b * α w a,b are perspective cover relations in Tam(α).
Proof. Let v ∈ S α (231) and let (a, b) ∈ Des(v). By Lemma 3.10, there is a unique u ∈ S α (231) with the desired properties.
Suppose that a is in the j th α-region.
Proof of Proposition 3.5. This follows from Proposition 3.11 using Lemma A.2 and Theorem A.10.
Moreover, for α = (1, 1, . . . , 1), the canonical join complex of Tam(α) (see Section A.4) was studied in [5]. In particular, it was shown in [5,Theorem 1.3] that this complex is vertex decomposable, a strong topological property introduced in [33] which implies that this complex is homotopic to a wedge of spheres, shellable and Cohen-Macaulay. We plan to investigate the canonical join complex of Tam(α) for arbitrary α in a follow-up article. For the time being, we pose the following conjecture. 3.3. The core label order of Tam(α). In this section we study the core label order of Tam(α), see Section A.3. By Proposition 3.1, Tam(α) is congruence uniform, and thus admits an edge-labeling with join-irreducible (α, 231)-avoiding permutations, which is determined by the perspectivity relation; see Lemma A.2 and Proposition 3.11.
The core label order of Tam(α) orders the elements of S α (231) with respect to this labeling. By Corollary 3.7, the elements of JoinIrr Tam(α) correspond bijectively to α-arcs. Therefore, we may identify the core label set of w ∈ S α (231) with a collection of α-arcs.
Since both Weak(S α ) and Tam(α) are congruence-uniform lattices, it makes sense to distinguish the corresponding core label sets. For u ∈ S α (231), we write Ψ L (u) for the core label set in Weak(S α ), and Ψ α (u) for the core label set in Tam(α).
We now show that for u ∈ S α (231), the core label set Ψ α (u) induces a noncrossing α-partition. To that end, we define then there is nothing to show. Otherwise, Theorem A.10 implies that Des(u) = (a 1 , b 1 ), (a 2 , b 2 ), . . . , (a t , b t ) = ∅. We denote by G the subgroup of S α generated by the transpositions corresponding to these descents.
Since Ψ α (u) = ∅, we may pick any w a,b ∈ Ψ α (u). By Lemma A.6, w a,b ∈ Ψ L (u). Since Weak(S α ) is a principal order ideal in Weak(S n ) and S n is a Coxeter group, [41, Theorem 2.10.5] thus implies that w a,b ∈ G and Inv(w a,b ) ⊆ Inv(u). Since w a,b ∈ G, we may write the transposition swapping a and b as a product of the generators of G. This implies a ∼ Φ α (u) b, and therefore w a,b ∈ X(u).
By definition, there exists a sequence of integers k 0 , k 1 , . . . , k t such that a = k 0 and b = k t and (k If t = 1, then (a, b) ∈ Des(u). By Proposition 3.5, w a,b is a canonical joinand of u, which implies w a,b ∈ Ψ α (u).
We now relate the core label order of Tam(α) to the refinement order on Nonc(α). Given two partitions P 1 , P 2 ∈ Π α , we say that P 1 refines P 2 if every block of P 1 is contained in some block of P 2 ; we write P 1 ≤ ref P 2 in that case.  = (p, 1, 1, . . . , q) for some integers p, q > 0.
We conclude this section with the observation that the core label order of Tam(α) is always a meet-semilattice. Recall the definition of the intersection property from Section A.3. Proof. Let w ∈ S α . If we denote the core label set of w in Weak(S n ) by Ψ L;n and the core label set of w in Weak(S α ) by Ψ L;α , then Ψ L;n (w) = Ψ L;α (w), because Weak(S α ) is principal order ideal of Weak(S n ) by Theorem 2.4.
For j ∈ JoinIrr Weak(S n ) , if j ∈ Ψ L;α (w), then j ≤ L w by Corollary A.3. This means that j ∈ JoinIrr Weak(S α ) . Thus, CLO Weak(S α ) is an order ideal of CLO Weak(S n ) .
By [36,Proposition 5.1] (see also [3,Section 4]), CLO Weak(S n ) is a lattice, which means that CLO Weak(S α ) is a meet-semilattice. Thus, by Theorem A.5, Weak(S α ) has the intersection property. Now, Proposition A.7 implies that any quotient lattice of Weak(S α ) has the intersection property. By Theorem 2.5, this is the case for Tam(α).
In a preliminary draft of this article, we claimed that the poset Nonc(α), ≤ ref is a meet-semilattice. A referee has provided the following counterexample.    Then, P 1 , P 2 ∈ Nonc(α), but their intersection is At the moment, we do not have anything meaningful to say about the posets Nonc(α), ≤ ref , except for the cases in which they coincide with CLO Tam(α) .

THE α-TAMARI LATTICES ARE TRIM
In this section, we prove that Tam(α) is trim for every composition α of n > 0. We first study the join-irreducible elements of Tam(α) in greater detail.

The poset of irreducibles of Tam(α).
Recall from Corollary 3.7 that the joinirreducible elements of Tam(α) are in bijection with the α-arcs. Moreover, Corollary 3.8 describes one-line notation and inversion sets of the join-irreducibles, and immediately implies the next result. We may now describe the restriction of the weak order to the set JoinIrr Tam(α) . See Figure 8 for an illustration.
Proof of Theorem 1.4. By Corollary 4.2 we conclude that for w a,b ≤ L w a ,b to hold, it is necessary that a and a belong to the same α-region. This accounts for the r − 1 connected components of Weak JoinIrr Tam(α) , because a can be chosen from any but the last α-region and there is a total of r α-regions. Now suppose that a lies in the j th α-region, which means that a takes any of the values {p j−1 +1, p j−1 +2, . . . , p j }. For any choice of a, we can pick some b ∈ {p j +1, p j +2, . . . , n} to obtain a join-irreducible element w a,b . Observe that whenever a = p j−1 +1, then w a,b ≤ L w a−1,b , and we always have w a,b ≤ L w a,b+1 when b < n. This implies that the j th component of Weak JoinIrr Tam(α) is isomorphic to the direct product of an α j -chain and an (α j+1 +α j+2 + · · · +α r )chain.
For α = (2, 1, 2), the maximal chain constructed in the proof of Proposition 4.4 is highlighted in Figure 6. We may now conclude the proof of Proposition 4.1.   (6). The labels appearing on C are pairwise distinct and they induce a total order on JoinIrr Tam(α) given by the following cover relations: if a = p j−1 + 1 and b = n, w p j+1 ,p j+1 +1 , if a = p j−1 + 1 and b = n, if a belongs to the j th α-region. Proof. With the notation from the proof of Proposition 4.4, the first p 1 · (n − p 1 ) cover relations along C are: α v (1,n−p 1 ) α · · · α v (2,1) α · · · α v (p 1 −1,n−p 1 ) . By construction, C is also a maximal chain in Weak(S α ) and it follows that (If k = n − p 1 , then we set k + 1 = 1.) The claim follows by induction.
Example 4.6. Let α = (2, 1, 2). The chain constructed in the proof of Proposition 4.4 is highlighted in Figure 6. The total order of the join-irreducibles of Tam (2, 1, 2) is This order corresponds to the so-called inversion order of the longest element w o ∈ S n with respect to the linear Coxeter element. It seems that this correspondence works in general, i.e. the order defined in Corollary 4.5 recovers the inversion order of the parabolic longest element w o;α ∈ S α with respect to the linear Coxeter element.
We conclude this section with the proof of Theorem 1.1.
Proof of Theorem 1.1. Tam(α) is congruence uniform by Proposition 3.1 and trim by Proposition 4.1.

The Galois graph of Tam(α). By Proposition 4.4, Tam(α)
is an extremal lattice. Any extremal lattice can be described in terms of a directed graph; its Galois graph, see Section A.6.
In this section, we give an explicit description of the Galois graph of Tam(α). We exploit the fact from Proposition 3.1 that Tam(α) is also congruence uniform.
Let us recall the following useful characterization of inversion sets of joins in the weak order.
and f (α) is defined in (1). There exists a directed edge s → t in Galois Tam(α) if s = t and j s ≤ m t , where the join-and meet-irreducible elements of Tam(α) are ordered as in (9). By Proposition 3.1, Tam(α) is also congruence uniform, so that Corollary A.18(ii) implies s → t if and only if s = t and j t ≤ j t * ∨ j s . We may thus view Galois Tam(α) as a directed graph on the vertex set JoinIrr Tam(α) . Now, pick w a,b , w a ,b ∈ JoinIrr Tam(α) such that w a,b = w a ,b and a belongs to the i th α-region and a belongs to the i th α-region. We need to characterize when (2) w a ,b ≤ w a ,b * ∨ α w a,b .
For simplicity, let us write w = w a,b , w = w a ,b and w * = w a ,b * . Let z = w * ∨ α w.
Let us first consider the case where a and a belong to the same α-region, i.e. i = i . There are two cases.
(i) Let a ≤ a . If b ≤ b, then Corollary 4.2 implies w ≤ L w and (2) holds. If b < b , then by Corollary 3.8, (a , b ) / ∈ Inv(w). In fact, (c, b ) / ∈ Inv(w) for any c ∈ [n], and if (a , c) ∈ Inv(w), then (2) is not satisfied. (ii) Let a > a . If b ≤ b , then Corollary 4.2 implies w < L w , so that (2) does not hold. If b > b , then by Corollary 3.8, (a , b ) / ∈ Inv(w). Again, (a , c) / ∈ Inv(w) for any c ∈ [n], and if (c, b ) ∈ Inv(w), then a ≤ c ≤ p i . However, if (a , c) ∈ Inv(w * ), (2) is not satisfied.  Let us now consider the case where a and a belong to different α-regions, i.e. i = i . By Corollary 3.8, (a , b ) / ∈ Inv(w). As before, we may actually conclude (a , c) / ∈ Inv(w) for all c ∈ [n], and if (c, b ) ∈ Inv(w), then a ≤ c ≤ p i and for any c ∈ [n] and (2) cannot be satisfied. If p i < b , then we may choose c = a to see that (a , b ) ∈ Inv(z) which implies (2). Figure 9 shows Galois Tam (1, 2, 1) and Galois Tam (2, 1, 2) . In [45, Theorem 5.5] it was shown that the complement of the undirected Galois graph of an extremal semidistributive lattice is precisely the 1-skeleton of the canonical join complex. By Proposition 3.5 the canonical join representations in Tam(α) correspond to noncrossing α-partitions. We thus have the following corollary (which may also be verified directly).   = (1, 1, . . . , 1).
We conclude the proof of Theorem 1.2.
Proof of Theorem 1.2. By Theorem 3.19, Tam(α) has the intersection property, whichby Theorem A.5-implies that CLO Tam(α) is a meet-semilattice. Clearly, a meet-semilattice is a lattice if and only if it has a greatest element. By [29,Lemma 4.6], the core label order of a congruence-uniform lattice L has a greatest element if and only if L is spherical. By Theorem 4.10, Tam(α) is spherical if and only if α = (n) or α = (1, 1, . . . , 1).
The motivation for the consideration of the M α -triangle comes from [13], where the corresponding polynomial for α = (1, 1, . . . , 1) was introduced. More precisely, F. Chapoton considered the generating function of the Möbius function of the noncrossing partition lattice with respect to the number of bumps. In view of Theorems 3.4 and 3.18, Chapoton's M-triangle agrees with our M (1,1,...,1) -triangle.
Computer experiments suggest the following conjecture.
Conjecture 5.3. Let n > 0 and let α be a composition of n into r parts. The rational functions H α (x, y) and F α (x, y) are polynomials with nonnegative integer coefficients if and only if α has at most one part exceeding 1.
If we replace the exponent r − 1 in the definition of H α (x, y) and F α (x, y) by Des(w) , then we can verify directly that (4) and (5) produce polynomials with integer coefficients; however these coefficients need not all be nonnegative, as can be witnessed in the example α = (2, 1, 2). If α has at most one part exceeding 1, then Conjecture 5.3 implies the existence of combinatorial families A H;α and A F;α , and combinatorial statistics σ 1 , σ 2 , τ 1 , τ 2 , such that We close by suggesting candidates for A H;α and σ 1 , σ 2 . For n > 0, we define and we set (i 1 , j 1 ) (i 2 , j 2 ) if and only if i 1 ≥ i 2 and j 1 ≤ j 2 .
Then, H α (x, y) =H α (x, y) if and only if α has at most one part exceeding 1.
Given a, b ∈ P with a ≤ b, the set [a, b] def = {c ∈ P | a ≤ c ≤ b} is an interval of P. Two elements a, b ∈ P form a cover relation if a < b and [a, b] consists of two elements. In that case, we usually write a b, and we say that a is covered by b and that b covers a. The set of cover relations of P is denoted by E (P).
An element a ∈ P is minimal (resp. maximal) in P if b ≤ a (resp. a ≤ b) implies b = a for all b ∈ P. If P has a unique minimal element (usually denoted by0) and a unique maximal element (usually denoted by1), then P is bounded. In a bounded poset, any element covering0 (resp. covered by1) is an atom (resp. coatom).
A chain (resp. antichain) of P is a subset of P in which every two distinct elements are comparable (resp. incomparable). A chain consisting of k elements is also called a k-chain. A chain is saturated if it can be written as a sequence of cover relations, and it is maximal if it is saturated and contains a minimal and a maximal element.
When M is a set, then a map f : E (P) → M is an edge-labeling of P. If C : a 0 a 1 · · · a s is a saturated chain, then we write a 1 ), f (a 1 , a 2 ), . . . , f (a s−1 , a s ) .
An order ideal (resp. order filter) of P is a subset I ⊆ P such that for all a ∈ I if b ≤ a (resp. a ≤ b) then b ∈ I. An order ideal (resp. order filter) is principal if it has a unique maximal (resp. minimal) element.
If for all a, b ∈ P there exists a least upper bound a ∨ b (resp. a greatest lower bound a ∧ b), then P is a join-semilattice (resp. meet-semilattice). If it exists, then a ∨ b (resp. a ∧ b) is the join (resp. the meet) of a and b. A poset that is both a join-and a meet-semilattice is a lattice. Every finite lattice is a bounded poset.
A.2. Congruence-uniform lattices. Let L = (L, ≤) be a finite lattice. A lattice congruence is an equivalence relation Θ on L such that for all a, b, The set Con(L) of all lattice congruences on L forms a distributive lattice under refinement [18]. If a b in L, then we denote by cg(a, b) the finest lattice congruence on L in which a and b are equivalent.
A consequence of Theorem A.1 is the existence of a surjective map cg * : JoinIrr(L) → JoinIrr Con(L) , j → cg(j).
A finite lattice is congruence uniform if cg * is a bijection for both L and L d . In that case, Theorem A.1 implies the existence of an edge-labeling (6) λ : E (L) → JoinIrr(L), (a, b) → j, where j is the unique join-irreducible element of L with cg(j) = cg(a, b).
The cover relations in a congruence-uniform lattice with the same label under λ can be characterized as follows. Two cover relations (a, b), In that case, we write (a, b) (c, d). See Figure 13 for an illustration. This lemma has the following consequences. Proof. By Lemma A.2, (a, b) (j * , j), which implies that either b ≤ j or j ≤ b. If b < j, then b ∨ j * = j, which implies that b ≤ j * . This means that there is a saturated chain from b to j which does not contain j * ; in particular j has at least two lower covers. This contradicts j ∈ JoinIrr(L). Proof. Let C : a 0 a 1 · · · a s be a saturated chain of L, and pick some index i ∈ [s] such that λ(a i−1 , a i ) = k ∈ JoinIrr(L). By Corollary A.3, k ≤ a i . Thus, for any j ≥ i it follows that k ≤ a j , and thus k ∨ a j = a j a j+1 . Hence, (k * , k) and (a j , a j+1 ) are not perspective, and thus λ(a j , a j+1 ) = k by Lemma A.2.
A.3. The core label order of a congruence-uniform lattice. The labeling (6) of a congruence-uniform lattice L = (L, ≤) gives rise to an alternate way of ordering L. This order was first considered by N. Reading in connection with posets of regions of simplicial hyperplane arrangements under the name shard intersection order; see [38,.
For a ∈ L, we define its nucleus by and we define the core label set of a by If no confusion can arise, we drop the subscript "L" from the core label set. The (a) A congruence-uniform lattice with edge labeling λ.
The core label order of the lattice from Figure 14a. We have represented the elements by their core label sets. Figure 14. The core label order of a congruence-uniform lattice.
which identifies the quotient lattice L/Θ of L by Θ with the restriction of L to the minimal elements in the congruence classes of Θ. Then, Lemma A.6 can be rephrased as: ([29,Proposition 4.11]). The intersection property is inherited by quotient lattices.
A.4. Semidistributive lattices. A finite lattice L = (L, ≤) is join semidistributive if for all a, b, c ∈ L with a ∨ b = a ∨ c follows a ∨ (b ∧ c) = a ∨ b. We may define meet-semidistributive lattices dually. A lattice is semidistributive if it is both join and meet semidistributive. The converse of Theorem A.8 is not true, see for instance [32,Section 3]. Joinsemidistributive lattices have another characteristic property: every element can be represented canonically as the join of a particular set of join-irreducible elements.
More precisely, a subset A ⊆ L is a join representation of a ∈ L if a = A. A join representation is irredundant if there is no proper subset of A that joins to a. For two irredundant join representations A 1 and A 2 of a we say that A 1 refines A 2 if for every a 1 ∈ A 1 there exists some a 2 ∈ A 2 with a 1 ≤ a 2 . In other words, the order ideal generated by A 1 is contained in the order ideal generated by A 2 . A join representation of a is canonical if it is irredundant and refines every other irredundant join representation of a. If a canonical join representation of a exists, then it is an antichain of join-irreducible elements; the canonical joinands of a.   Proposition 2.2 in [37] states that in any finite lattice L, every subset of a canonical join representation is again a canonical join representation. Thus, the set of canonical join representations of L forms a simplicial complex; the canonical join complex of L. If L is join semidistributive, then the faces of this complex are indexed by the elements of L. See [4] for more information on the canonical join complex.
If L is congruence uniform, then we can use the labeling from (6)  Proof. Let a ∈ L. Let b 1 , b 2 ∈ L be such that b 1 = b 2 and λ(b 1 , a) = k = λ(b 2 , a). By Corollary A.3, k ≤ a, and by Lemma A.2, The claim now follows from Theorem A.10.
Proposition A.12 ([6,Proposition 4.11]). Let L be a finite, join-semidistributive lattice and let Θ be a lattice congruence on L. If j is a canonical joinand of a ∈ L such that j is not contracted by Θ, then j is a canonical joinand of π ↓ Θ (a) in L. Moreover, if π ↓ Θ (a) = a, then none of the canonical joinands of a is contracted by Θ.
Let j ∈ JoinIrr(L). If the set {a ∈ L | j * ≤ a, j ≤ a} has a greatest element, then we denote it by κ(j). Whenever κ(j) exists, it must be meet irreducible. We recall two facts about the partial map κ : JoinIrr(L) → MeetIrr(L). Lemma A.14 ([17, Lemma 2.57]). Let L = (L, ≤) be a finite lattice, and let j ∈ JoinIrr(L) be such that κ(j) exists. For every a ∈ L we have a ≤ κ(j) if and only if j ≤ j * ∨ a.
A.5. Trim Lattices. Let L = (L, ≤) be a finite lattice. The length of a chain of L is one less than its cardinality. Let (L) denote the maximum length of a maximal chain of L. For every finite lattice, the following holds: (8) (L) ≤ min JoinIrr(L) , MeetIrr(L) .
If these three quantities are the same, i.e. if JoinIrr(L) = (L) = MeetIrr(L) , then L is extremal [26]. It follows from [26,Theorem 14(ii)] that any finite lattice can be embedded as an interval into an extremal lattice. Consequently, extremality is not inherited by intervals.
In [44], a strengthening of extremality was introduced which does have this hereditary property. An element a ∈ L is left modular if for all b, c ∈ L with b < c it holds that If L has a maximal chain of length (L) consisting entirely of left-modular elements, then L is left modular. An extremal, left-modular lattice is trim [44].
It was recently shown that any extremal, semidistributive lattice is already trim.  Figure 15 shows the smallest extremal lattice that is not left modular. It has only one chain of maximum length, but the non-atom marked in green is not left modular.
A.6. The Galois graph of an extremal lattice. Extremal lattices can be compactly represented in terms of a directed graph-the Galois graph-which encodes the incomparability relation between join-and meet-irreducible elements [26,Theorem 11].
Let L = (L, ≤) be extremal with (L) = n, and fix a maximal chain C :0 = a 0 a 1 · · · a n =1. Then, JoinIrr(L) = n = MeetIrr(L) . We can label the join-irreducible elements by j 1 , j 2 , . . . , j n and the meet-irreducible elements by m 1 , m 2 , . . . , m n such that (9) j 1 ∨ j 2 ∨ · · · ∨ j s = a s = m s+1 ∧ m s+2 ∧ · · · ∧ m n for all s. We can always order some of the irreducibles in such a way; the extremality guarantees that this is an ordering of all irreducibles. Using this order, we define the Galois graph of L following [26, Theorem 2(b)] (see also [45,Section 2.3]). This is the directed graph Galois(L) with vertex set [n], where s → t if and only if s = t and j s ≤ m t . Figure 16b shows the Galois graph of the extremal lattice in Figure 16a.
The Galois graph of an extremal lattice L uniquely determines L as we will briefly outline next. In general, let G = [n], E be a directed simple graph. An orthogonal pair of G is a pair (A, B) with A, B ⊆ [n], A ∩ B = ∅ and there is no (s, t) ∈ E with s ∈ A, t ∈ B. An orthogonal pair is maximal if both sets A and B are (cardinality-wise) maximal with that property. We may define a partial order on the set of maximal orthogonal pairs of G by setting (A 1 , B 1 ) (A 2 , B 2 ) if and only if A 1 ⊆ A 2 (or equivalently B 2 ⊆ B 1 ). The set of maximal orthogonal pairs of G with respect to this partial order is a lattice. Extremal lattices may now be characterized via this construction; see also [45,Section 2.3].
Theorem A.16 ([26,Theorem 11]). Every finite extremal lattice is isomorphic to the lattice of maximal orthogonal pairs of its Galois graph. Conversely, given any directed     graph G = [n], E such that (s, t) ∈ E only if s > t, the lattice of maximal orthogonal pairs is extremal. Figure 17 shows two lattices of maximal orthogonal pairs. Constructing the Galois graph of an extremal lattice requires to understand the incomparability relation between join-and meet-irreducible elements. If an extremal lattice is additionally congruence uniform, we may simplify this construction by only taking the join-irreducible elements into account.
If L is both extremal and congruence uniform, then we may define another ordering of the join-irreducible elements using the labeling λ from (6). In particular, we pick a maximal chain of maximum length, and we order the join-irreducible elements according to the order in which they appear in λ(C). This is a total order of all join-irreducible elements of L by Corollary A.4 and the assumption that L is extremal.
Lemma A.17. Let L be a finite, extremal and congruence-uniform lattice, and fix a maximal chain C of maximum length. The ordering of JoinIrr(L) coming from (9) agrees with the order in which the join-irreducible elements appear in λ(C).
Proof. Let (L) = n, and pick a maximal chain C :0 = a 0 a 1 · · · a n =1. Suppose that-with respect to C-the order of the join-irreducible elements of L from (9) is j 1 , j 2 , . . . , j n .
Since (9) determines a linear order on JoinIrr(L), it follows that j = j t . The claim follows by induction.
Corollary A.18. Let L be a finite, extremal and congruence-uniform lattice, in which JoinIrr(L) and MeetIrr(L) are ordered as in (9) with respect to some maximal chain of length n = (L).
(ii) For s, t ∈ [n], j s ≤ m t if and only if s = t and j t ≤ j t * ∨ j s .
(iii) If j t ≤ j s , then there is a directed edge from s to t in Galois(L).
Proof. (i) Let (L) = n, and let s ∈ [n]. By Lemma A.13, there exists m = κ(j s ). By definition of κ, j s * ≤ m and j s ≤ m and m is maximal with this property. Thus j s ∧ m = j s * and j s ∨ m = m * . Thus (j s * , j s ) (m, m * ), and Lemma A.2 implies λ(j s * , j s ) = λ(m, m * ). Let C :0 = a 0 a 1 · · · a n =1. By Lemma A.17, λ(a s−1 , a s ) = λ(j s * , j s ) = λ(m, m * ), and applying Lemma A.2 once more yields (a s−1 , a s ) (m, m * ). Since m is meet irreducible, we conclude-by the dual statement of Corollary A.3-that a s−1 = m ∧ a s . By (9), m = m s . (ii) This follows from Lemma A.14 and (i).
(iii) This follows from (ii) and the definition of Galois(L).
If L is extremal and congruence uniform, then Corollary A.18 implies that we may view Galois(L) as a directed graph with vertex set JoinIrr(L) where we have a directed edge j s → j t if and only if s = t and j t ≤ j t * ∨ j s . For extremal lattices that are not congruence uniform, this construction normally yields a directed graph that is not isomorphic to Galois(L). This is illustrated in Figure 16. Figure 16c shows the directed graph with vertex set [4] and a directed edge s → t if and only if s = t and j t ≤ j t * ∨ j s holds in the extremal lattice in Figure 16a.
A.7. Poset topology. The order complex ∆(P) of a finite poset P = (P, ≤) is the simplicial complex whose faces are the chains of P. If P has a least or a greatest element, then ∆(P) is always contractible. If P is bounded, then we denote by P def = P \ {0,1}, ≤) the proper part of P. The Möbius function of P is the map µ P : P × P → Z, inductively defined by µ P (a, a) def = 1 for all a ∈ P and by µ P (a, b) def = − ∑ c∈P : a≤c<b µ P (a, c), for all a, b ∈ P with a = b. If P is bounded, then the Möbius number of P is µ(P) def = µ P (0,1).
It follows from a result of P. Hall that the Möbius number of P equals the reduced Euler characteristic of ∆(P); see [40,Proposition 3.8.5].
Let us recall two results concerning the Möbius number of certain kinds of lattices. The first one follows from [25] (see also [28,Theorem 8]) and [9, Theorem 5.9].
Theorem A. 19. Let L be a finite, left-modular lattice. The order complex ∆(L) is homotopic to a wedge of µ(L) -many spheres.