Tamari Lattices for Parabolic Quotients of the Symmetric Group

. We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231 -avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231 -avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups.


Introduction
The Tamari lattice T n was introduced by D. Tamari as a partial order on binary bracketings of a string of length n + 1 [13]. The number of such bracketings is given by the nth Catalan number Cat(n) := 1 n+1 2n n . A. Björner and M. Wachs observed in [3,Section 9] that T n is the sublattice of the weak order on S n consisting of the subset S n (231) of permutations whose inversions sets satisfy a certain "compressed" condition (the notation reflects that these permutations are more commonly refered to as the 231-avoiding permutations). This observation was the precursor to N. Reading's definition of the Cambrian lattices, which may be described as the restriction of the weak order of a finite Coxeter group to certain aligned elements, which are also characterized via their inversion sets [8]. The Cambrian lattices thus naturally generalize the Tamari lattice to any finite Coxeter group and any Coxeter element.
In this abstract, we propose a new generalization of the Tamari lattice to parabolic quotients S J n of the symmetric group by introducing a distinguished subset of permutations, denoted by S J n (231), that play the role of the 231-avoiding permutations for the parabolic quotient. We characterize these permutations both in terms of a generalized notion of pattern avoidance, and-in our opinion, more naturally-in terms of their inversion sets.
Theorem 1 Let S denote the set of simple transpositions of S n for n > 0. For J ⊆ S, the restriction of the weak order to S J n (231) forms a lattice, which we call T J n -the Tamari lattice for the parabolic quotient S J n . Furthermore, T J n is a lattice quotient of the weak order on S J n . When J = ∅, we recover the Tamari lattice on 231-avoiding permutations. In the remainder of this article, we refer to T J n as the parabolic Tamari latttice. (i) We recall the necessary definitions for parabolic quotients of the symmetric group in Section 2 and prove Theorem 1 in Section 3.
N. Reading's aligned elements of a Coxeter group W have a markedly different description as the sortable elements, generalizing the result that 231-avoiding permutations are precisely those permutations that are stack-sortable [5, Exercise 2.2.1. [4][5]. For any Coxeter element, these sortable elements provide a bridge between two famous families of Coxeter-Catalan objects, namely W -noncrossing partitions and W -clusters [9]. Remarkably, these objects are enumerated uniformly by a simple product formula depending on the degrees of W [12, Remark 2]. Although we no longer have a product formula for |S J n (231)|, we are able to generalize certain other Catalan objects to the parabolic setting. In Sections 4 and 5 we generalize noncrossing and nonnesting partitions to parabolic quotients as the sets NC J n and NN J n , and we prove bijectively that our three parabolic generalizations are equinumerous. Theorem 2 For n > 0 and J ⊆ S, we have Finally, we propose a generalization of T J n to any element of any Coxeter group in Section 6.

The Symmetric Group
In this section, we recall the definitions of weak order, 231-avoiding permutations, and parabolic quotients.

Weak Order
The symmetric group S n is the group of permutations of [n] := {1, 2, . . . , n}. Let S := {s 1 , s 2 , . . . , s n−1 } denote the set of simple transpositions of S n , i.e. s i := (i i+1) for i ∈ [n − 1]. It is well known that S n is isomorphic to the Coxeter group A n−1 , and so admits a presentation of the form Although this abstract focuses on constructions for S n , many of the definitions may be generalized to Coxeter groups (see the discussion in Section 6).
We may specify permutations using their one-line notation: The weak order Weak(S n ) is the partial order defined by u ≤ S v if and only if inv(u) ⊆ inv(v). We refer the reader to [2, Section 3] for more background on the weak order, in particular in the broader context of Coxeter groups. A remarkable property of Weak(S n ) is that it is a lattice [2, Theorem 3.2.1]. Recall further that there is a unique maximal element w • in Weak(S n ) whose one-line notation is w • = n(n − 1) · · · 1. We denote the identity permutation by e.
A permutation w ∈ S n is called 231-avoiding if there exists no triple i < j < k such that w k < w i < w j . Let S n (231) denote the set of 231-avoiding permutations of S n . These elements may also be specified by a "compressed" condition on their inversion sets-an inversion set inv(w) is compressed if i < j < k and (i, k) ∈ inv(w) implies that (i, j) ∈ inv(w). Then w ∈ S n (231) if and only if inv(w) is compressed [3,Section 9]. We recall the following result due to A. Björner and M. Wachs, which will serve as a definition of the Tamari lattice T n for our purposes.

Parabolic Quotients
Recall that any subset J ⊆ S generates a subgroup of S n isomorphic to a direct product of symmetric groups of smaller rank. We call such a subgroup parabolic, and we denote it by (S n ) J . Moreover, we define the parabolic quotient of S n with respect to J by The set S J n thus consists of the minimal length representatives of the right cosets of the corresponding parabolic subgroup. By [2,Proposition 2.4.4], any permutation w ∈ S n can be uniquely written as w = w J · w J , for w J ∈ S J n and w J ∈ (S n ) J . We further define the parabolic weak order Weak(S J n ) by the cover relations ws J S w if and only if ws ∈ S J n for some s ∈ S. Weak(S J n ) has a longest element w J • , is isomorphic to the weak order interval [e, w J • ], and is therefore a lattice. In particular this allows us to use ≤ S instead of ≤ J S .

Tamari Lattices for Parabolic Quotients of S n
In this section, we define S J n (231)-a generalization of the set of 231-avoiding permutations to parabolic quotients of the symmetric group. We characterize the inversion sets of the permutations in S J n (231) and prove that (S J n (231), ≤ S ) is a lattice. We call the parts of B(J) the J-regions. We indicate the parts of B(J) in the one-line notation of a permutation w ∈ S J n by vertical bars. Lemma 4 If w ∈ S J n , then the one-line notation of w has the form w = w 1 < · · · < w j1 | w j1+1 < · · · < w j2 | · · · | w jr+1 < · · · < w n .
Definition 5 A permutation w ∈ S J n is J-231-avoiding if there exist no three indices i < j < k, all of which lie in different J-regions, such that w k < w i < w j and w i = w k + 1. Let S J n (231) denote the set of J-231-avoiding permutations of S J n .

Example 6
The left image in Figure 1 shows Weak S In the case J = ∅, the set of J-231-avoiding permutations is equal to the set of 231-avoiding permutations.
Proposition 7 If w ∈ S n has a 231-pattern, then there exist indices i < j < k such that w k < w i < w j and w i = w k + 1. Consequently, S ∅ n (231) = S n (231) We now generalize A. Björner and M. Wachs' definition of compressed inversion sets to parabolic quotients. We remark that the compressed inversion sets are a special case of N. Reading's aligned condition, which generalizes the notion to any finite Coxeter group and any Coxeter element.

Tamari Lattices for Parabolic Quotients of S J n
Now we prove the first part of Theorem 1, namely that S J n (231), ≤ S is a lattice. The proof follows from the next lemma, which is modeled after [8, Lemma 5.6], a result that covers the case J = ∅.
Lemma 10 For every w ∈ S J n , there exists some w ∈ S J n such that inv(w ) is the unique maximal set under containment among all J-compressed inversion sets inv(u) ⊆ inv(w). In other words, for every w ∈ S J n , there exists a unique maximal J-231-avoiding permutation w with w ≤ S w.
Proof: We pick w ∈ S J n , and proceed by induction on the cardinality of inv(w). If inv(w) = ∅, then inv(w) is J-compressed and the claim holds trivially. Suppose that |inv(w)| = r, and that the claim is true for all x ∈ S J n with |inv(x)| < r. If inv(w) is already J-compressed, then we set w = w and we are done. Otherwise, Lemma 9 implies that w contains an instance of a J-231-pattern, which means that there are indices i < j < k that all lie in different J-regions such that w k < w i < w j and w i = w k + 1. Consider the lower cover u of w that has w i and w k exchanged, i.e. u = s w k w. In particular, we have inv(w) = inv(u) ∪ (i, k) . By induction hypothesis, there exists some u ∈ S J n such that inv(u ) is the unique maximal J-compressed inversion set that is contained in inv(u). We claim that w = u .
In order to prove this claim, we pick some element . We assume the opposite, and in view of the previous reasoning it follows that v i > v k + 1. Let d be the index such that v d = v k + 1, and let e be the index such that v i = v e + 1. Since w i = w k + 1, we observe: We have the following relations: which is a contradiction.) We now distinguish five cases: . It follows that w i > w d and w d > w k , which contradicts (D).
(iv) Assume i < k < e. Hence (i, k), (i, e) ∈ inv(v) ⊆ inv(w). It follows that w i > w e , and (E) implies w e < w k . Lemma 4 implies that k and e lie in different J-regions.
. It follows that w i > w d as well as w e > w k . Now (D) and (E) imply w d < w k and w e > w i , respectively. Lemma 4 implies that e, i, k and d all lie in different J-regions.
Let e be the smallest element in the J-region of e such that v e > v d , and let d be the largest element in the J-region of d such that v d < v e . We record that e ≤ e < i < j < k < d ≤ d , and we proceed by induction on If f < i and they do not lie in the same J-region, then we can consider the triple (f, i, d ), and we obtain a contradiction by induction, f > i and they do not lie in the same J-region, then we can consider the triple (e , i, f ), and we obtain a contradiction by induction, since v e − v f < v e − v d . If f and i lie in the same J-region, then we can consider the triple (f, j, d ), and we obtain a contradiction by induction, since v f − v d < v e − v d . (Note that we can prove the induction base verbatim using v j instead of v i .) We have thus shown that (i, k) / ∈ inv(v), which implies inv(v) ⊆ inv(u). By induction assumption it follows that inv(v) ⊆ inv(u ), which proves w = u . The reformulation of this statement follows from Lemma 9. 2

Proposition 11
The poset S J n (231), ≤ S is a lattice.
Proof: Let w 1 , w 2 ∈ S J n (231), and let u = w 1 ∧ w 2 denote their meet in weak order. Lemma 10 implies that there exists a unique maximal element u ∈ S J n (231) with u ≤ S w 1 , w 2 , which then necessarily must be the meet of w 1 and w 2 in S J n (231), ≤ S . Since S J n (231), ≤ S is a finite meet-semilattice with a greatest element (w J • is trivially J-231-avoiding), it is a classical lattice-theoretic result that it must be a lattice. 2 Lemma 7 implies that the set S ∅ n (231) coincides with the set of all classical 231-avoiding permutations of S n , and Theorem 3 states that S ∅ n (231), ≤ S is isomorphic to the Tamari lattice T n . In view of Proposition 11, we feel it is justified to denote the poset S J n (231), ≤ S by T J n , and call it the parabolic Tamari lattice. is 2|14|3. In certain special cases, for example when J = ∅ or when J = S \ {s}, we do obtain sublattices.

Parabolic Tamari Lattices are Lattice Quotients
In this section, we prove that T J n is a lattice quotient of the weak order interval [e, w J • ], completing the proof of Theorem 1. Recall from [8, Section 3] that an equivalence relation Θ on a lattice (L, ≤) is a lattice congruence if and only if all equivalence classes [x] Θ are intervals, and the projections that map x to the least or greatest element in [x] Θ , respectively, are both order-preserving.
By Lemma 10, we obtain a projection Π J ↓ : S J n → S J n (231) by Π J ↓ (w) = w , where w is the unique maximal J-231-avoiding permutation below w.

Lemma 14 The fibers of Π
We now define a set of permutations "dual" to the J-231-avoiding permutations (ii) . We say a permutation w ∈ S J n is J-132-avoiding if there are no three indices i < j < k, all of which lie in different J-regions, such that w i < w k < w j and w k = w i + 1. We can prove the following result analogously to Lemma 10.
Lemma 15 For any w ∈ S J n , there is a unique minimal J-132-avoiding permutation w with w ≤ S w .
Hence we obtain a map Π J ↑ : S n → S J n (132) that maps w to the unique minimal J-132-avoiding permutation w above w.
Lemma 16 The maps Π J ↓ and Π J ↑ are order-preserving.
Lemma 17 Let u, v ∈ S J n with u S v. The following are equivalent: -Lemma 17 implies that there must be a J-231 pattern in v that is not present in u. In other words there are indices i < j < k, all of which lie in different J-regions with v k < v i < v j and v i = v k + 1, and u is obtained from v by removing the inversion (i, k). Hence we have u i < u k < u j and u k = u i +1, which forms a J-132-pattern in u. We is minimal among all J-132-avoiding permutations above v, we obtain the desired equality. If u < S v do not form a cover relation, we find the desired equality by repeated application of the previous reasoning.
Finally, suppose that u and v are incomparable. We conclude that Π J ↓ (u ∧ v) = Π J ↓ (u), since Π J ↓ (u) is the unique maximal J-231-avoiding permutation below both u and v. In view of the reasoning in the first paragraph, we find Π J

Parabolic Noncrossing Partitions
In this section, we define noncrossing partitions for parabolic quotients NC J n and we give a bijection between NC J n and S J n (231). A set partition of [n] is a collection P = {P 1 , P 2 , . . . , P s } of subsets of [n] with the property that P i ∩ P j = ∅ for 1 ≤ i < j ≤ s, and s i=1 P i = {1, 2, . . . , n}. The elements P i of P are called the parts of P . A pair (a, b) is a bump of P if a, b ∈ P i for some i ∈ {1, 2, . . . , s} and there is no c ∈ P i with a < c < b.
Definition 20 A partition P of [n] is J-noncrossing if it satisfies the following three conditions: (NC1) If i and j lie in the same J-region, then they are not contained in the same part of P.
(NC2) If two distinct bumps (i 1 , i 2 ) and (j 1 , j 2 ) of P satisfy i 1 < j 1 < i 2 < j 2 , then either i 1 and j 1 lie in the same J-region or i 2 and j 1 lie in the same J-region.
We denote the set of all J-noncrossing set partitions of [n] by NC J n . If J = ∅, then we recover the classical noncrossing set partitions. We now introduce a combinatorial model for the J-noncrossing partitions. We draw n dots, labeled by the numbers 1, 2, . . . , n, on a straight line, and highlight the J-regions by grouping the corresponding dots together. For any bump (i, j) in P ∈ NC J n , we draw an arc connecting the dots corresponding to i and j, respectively, that passes below all dots corresponding to indices k > i that lie in the same J-region as i, and above all other dots between i and j. See the bottom left of Figure 2 for an illustration.
Using this combinatorial model for NC J n , we now prove the following theorem. Theorem 21 For n > 0 and J ⊆ S, we have S J n (231) = NC J n .
Proof: Let w ∈ S J n . We construct a set partition P of {1, 2, . . . , n} by associating a bump (i, j) with any inversion (i, j) ∈ inv(w) satisfying w i = w j + 1. If (i, j) is a bump of P, then it follows from Lemma 4 that i and j lie in different J-regions, and therefore condition (NC1) is satisfied. Suppose (i 1 , i 2 ) and (j 1 , j 2 ) are two different bumps of P with i 1 < j 1 < i 2 < j 2 , but neither i 1 , j 1 nor i 2 , j 1 are in the same block. If w i1 < w j1 , then (i 1 , j 1 , i 2 ) induces a J-231-pattern in w, which is a contradiction. If w i1 > w j1 , it follows that w j1 < w i2 , and then (j 1 , i 2 , j 2 ) induces a J-231-pattern in w, which is a contradiction. Hence (NC2) is satisfied. Finally, suppose that (i 1 , i 2 ) and (j 1 , j 2 ) are two different bumps of P with i 1 < j 1 < j 2 < i 2 such that i 1 and j 1 are in the same J-region. Lemma 4 implies w i1 < w j1 . It follows that (i 1 , j 1 , i 2 ) induces a J-231-pattern in w, which is a contradiction. Hence (NC3) is satisfied, and so P ∈ NC J n . Conversely, let P ∈ NC J n . We construct a permutation w ∈ S J n (231) where every bump (i, j) of P corresponds to an inversion (i, j) ∈ inv(w) with w i = w j + 1. We proceed by induction on n. The case n = 1 is trivial. Suppose that for any n < n we can construct a J -231-avoiding permutation of S J n from a given J -noncrossing set partition of [n ], where J is the restriction of J to [n ]. That is, two entries i, j ∈ [n ] lie in the same J -region if and only if they lie in the same J-region.
LetP be the unique part of P such that 1 ∈P , and writeP = {i 1 , i 2 , . . . , i r }, i.e. i 1 = 1. It follows by construction that w i1 = w i2 + 1 = · · · = w ir + (r − 1). Now we compute the smallest possible value of w i1 . Let i k−1 < j < i k for some k ∈ {2, 3, . . . , r} lie in the same J-region as i k . Since we want to obtain a permutation in S J n (231), Lemma 4 forces w j < w i k . Suppose there are t 1 such indices. Now let i k−1 < j < i k for some k ∈ {2, 3 . . . , r} lie in a different J-region than i k . If w j > w i k , then (i k−1 , j, i k ) induces a J-231-pattern in w. This forces w j < w i k . Suppose there are t 2 such indices. It follows that the smallest value for w i1 is r + t 1 + t 2 . (Put another way, we count the vertices lying below any arc ofP , as well as the vertices lying below any arc that starts in the same J-region, but to the left of some element inP (including endpoints).) Now we removeP from P, and we create two smaller partitions from the remaining parts. The indices that contribute to the computation of w i1 above are put into a left partition P l , and the remaining indices are put into a right partition P r , where we keep all bumps. Both P l and P r can be seen as parabolic noncrossing set partitions of [n l ] and [n r ], respectively, where n l , n r < n. By induction we can create J-231-avoiding permutations w (l) and w (r) from these partitions. Now we obtain the value w j for j / ∈P as follows: If j ∈ P l , then w j = w (l) j if j is the j -th largest value in P l . If j ∈ P r , then w j = w Since all bumps in P occur only between elements inP , in P l or P r , it follows that w ∈ S J n (231). 2 Example 22 Let J = {s 1 , s 2 , s 3 , s 5 , s 8 }. Consider P ∈ NC J 10 given by the bumps (2,9), (3,10), (6,8). Since no bump starts in 1, we obtain w 1 = 1, and the corresponding right partition is the restriction of P to {2, 3, . . . , 10}. Here we haveP = {2, 9}, and the elements below the arc (2, 9) are 2, 5, 6, 7, 8, 9, (this is because 3 and 4 lie in the same J-region as 2, and the arc (2, 9) thus passes below 3 and 4). Hence we obtain w 2 = 7 and w 9 = 6. The corresponding left partition is P l = {5}, {6, 8}, {7} and the corresponding right partition is P r = {3, 10}, {4} . By induction, we conclude that w (l) = 13|4|2 and w (r) = 23|1. We fashion them together to form the permutation w = 1 7 9 10 | 2 4 | 5 | 3 6 | 8, which is indeed contained in S J 10 (231). By construction, { (2,9), (3,10), (6,8)} are those inversions in inv(w) whose values differ by 1, and are precisely the bumps of P.
Remark 23 It is not hard to check that the bijection of Theorem 21 is identical to the bijection given in [11] when restricted to the J-231-sortable elements, although it was discovered independently [14].

Parabolic Nonnesting Partitions
Another important special case of set partitions of [n] is that of nonnesting set partitions, which do not contain two bumps (i 1 , i 2 ) and (j 1 , j 2 ) such that i 1 < j 1 < j 2 < i 2 . These were introduced by A. Postnikov uniformly for all crystallographic Coxeter groups to be antichains in the corresponding root poset [12,Remark 2]. It turns out that (for any crystallographic Coxeter group) there are the same number of noncrossing and nonnesting partitions. Moreover, they are also equidistributed by part size, see [1].
In this section, we introduce a generalization of nonnesting set partitions to parabolic quotients of the symmetric group. We say that a partition P of [n] is J-nonnesting if it satisfies the following two conditions: (NN1) If i and j lie in the same J-region, then they are not contained in the same part of P.
(NN2) If (i 1 , i 2 ) and (j 1 , j 2 ) are two distinct bumps of P, then it is not the case that i 1 < j 1 < j 2 < i 2 .
Example 25 Consider the J-nonnesting set partition of [10] shown at the top left of Figure 2 indicated by the dark gray region. The construction of the smaller parabolic noncrossing partitions is shown in the middle and right part of that figure, and the resulting J-noncrossing partition is shown at the bottom left.

Generalizations
Let W be a Coxeter group with geometric realization in V so that the elements of W correspond to the regions of V −R, where R is the set of hyperplanes corresponding to the reflections of W . N. Reading has defined a method for slicing hyperplanes into pieces, which he calls shards [10]-these pieces correspond naturally to join-irreducible elements, and so all that follows can be rephrased using join-irreducibles and canonical join representations, or inversion sets and covered reflections. We can geometrically rephrase the construction of the J-231-sortable elements in an attractive way using N. Reading's shards. Fix c = s 1 s 2 · · · s n in type A n , let w J • (c) be the c-sorting word for w J • , and let R w J • (c) be the set of shards crossed by the gallery corresponding to w J • (c) (in other words, for each hyperplane in the inversion set of w J • , w J • (c) selects a connected piece of that hyperplane). Then the J-231-sortable elements are exactly those elements which are minimal in the regions V − R w J • (c) . This suggests the following definition for any Coxeter group W , any element w ∈ W , and any reduced word w for w.
Definition 26 Fix a Coxeter group W , an element w ∈ W , and a reduced word w for w. Let R w be the set of shards crossed by the gallery corresponding to w. An element u ∈ [e, w] is called w-sortable if and only if every lower facet of u lies on a shard in R w .
Although it is tempting to conjecture that the poset defined as the restriction of weak order on [e, w] to the w-sortable elements is always a lattice, this statement turns out to be false. For example, take w = s 2 s 1 s 2 s 3 s 4 s 2 s 3 s 1 s 2 s 1 in A 4 (we do not know of any counterexamples in rank less than 4). We have been unable to determine necessary and sufficient conditions on W and w for the resulting poset to be a lattice, although taking w to be a c-sorting word seems like a reasonable condition to consider further.
Restricting Definition 26 to c-sorting reduced words for longest elements of parabolic quotients w J • (c) generalizes the construction of this abstract to any finite Coxeter group and any Coxeter element. We do not know if the resulting posets are in general lattices, but many features do generalize, including the definitions of noncrossing and nonnesting partitions (although they are not always equinumerous) [14]. For example, we have the following analogue to the cluster complex using work of C. Ceballos, J.-P. Labbé, and C. Stump [4] and V. Pilaud and C. Stump [6].
Definition 27 Fix a finite Coxeter group W , a Coxeter element c and a parabolic quotient J. Define the (non-reduced) word in simple generators Q = cw • (c) and consider the set S c (W J ) of subwords of Q whose complements contain a reduced word for w J • . When J = ∅, the subword complex S c (W J ) is isomorphic to the c-cluster complex, and it is known that the flip graph of S c (W J ) is isomorphic to the restriction of weak order to the c-sortable elements [4]. We conjecture that the flip graph S c (W J ) is isomorphic to the restiction of [e, w J • ] to the w J • (c)-elements [14], and we are interested in the extent to which Definitions 26 and 27 are linked.