Parabolic Tamari Lattices in Linear Type B

We study parabolic aligned elements associated with the type-$B$ Coxeter group and the so-called linear Coxeter element. These elements were introduced algebraically in (M\"uhle and Williams, 2019) for parabolic quotients of finite Coxeter groups and were characterized by a certain forcing condition on inversions. We focus on the type-$B$ case and give a combinatorial model for these elements in terms of pattern avoidance. Moreover, we describe an equivalence relation on parabolic quotients of the type-$B$ Coxeter group whose equivalence classes are indexed by the aligned elements. We prove that this equivalence relation extends to a congruence relation for the weak order. The resulting quotient lattice is the type-$B$ analogue of the parabolic Tamari lattice introduced for type $A$ in (M\"uhle and Williams, 2019). These lattices have not appeared in the literature before.


INTRODUCTION
Knuth introduced in [17, Section 2.2.1] the family of stack-sortable permutations.These are permutations that can be converted into the identity permutation by passing through a stack.Combinatorially, stack-sortable permutations are characterized by avoiding the pattern 231.More precisely, a permutation π of {1, 2, . . ., n} avoids the pattern 231 if and only if for every i < j < k, when π(i) > π(k), we have π(i) > π(j).Following Bj örner and Wachs this pattern-avoidance can be interpreted as a certain forcing of inversions [4, Section 9].
Using a root-theoretic approach, Reading generalized this condition to all finite irreducible Coxeter groups W and all Coxeter elements c.These c-aligned elements of W play an important role in the very active stream of Coxeter-Catalan combinatorics.First, the number of c-aligned elements of W is the W-Catalan number (for any choice of Coxeter element c).Second, the c-aligned elements provide a bijective bridge between c-noncrossing partitions and c-clusters associated with W [27].
On top of that, the c-aligned elements behave nicely from a lattice-theoretic point of view.Inside the weak order, they form the c-Cambrian lattice; a particular semidistributive and trim lattice.This family of lattices generalizes the famous Tamari lattice [26].Another remarkable feature is that these lattices arise from a certain orientation of the 1-skeleton of the c-associahedron, a simple polytope associated with the Coxeter group W. In fact, the boundary complex of the dual polytope of the c-associahedron is the c-cluster complex associated with W, which arises from the almost positive roots associated with W by means of a certain compatibility condition.
In [23], c-aligned elements, c-noncrossing partitions and c-clusters were generalized to parabolic quotients of finite irreducible Coxeter groups.In this generalization, some properties of these objects were conjectured to remain, such as the lattice property of parabolic c-aligned elements under weak order ([23, Conjecture 35]), and-when W is of coincidental type-the equinumerosity of parabolic caligned elements, parabolic c-noncrossing partitions and parabolic c-clusters ([23,Conjecture 41]).The case of the symmetric group with c the increasing long cycle was settled in the same article.Further research has exhibited remarkable connections between parabolic Coxeter-Catalan objects associated with the symmetric group, certain Hopf algebras, and the theory of diagonal harmonics [6].For more properties of these objects, readers are referred to [10,18,22].
In this article, we present a first study of parabolic Coxeter-Catalan objects associated with the hyperoctahedral group, i.e., the Coxeter group of type B. We start by recalling the necessary basics on posets and lattices (Section 2) and Coxeter groups (Section 3).Then we turn our attention to the Coxeter group of type B, which can be viewed as the symmetry group of the hyperoctahedron.We recall a permutation representation of this group in terms of sign-symmetric permutations (Section 4.1), and then consider a particular realization of the roots associated with this group (Section 4.2).With the basics set up we then recall Reading's construction of the c-aligned elements associated with the Coxeter group of type B, where c is a linear Coxeter element, i.e., a particular long cycle (Section 4.3).
In the main part of this article, we consider parabolic quotients of the hyperoctahedral group of degree n.These parabolic quotients consist of minimal-length representatives of the left cosets with respect to certain subgroups.We give a simple criterion for the membership of a sign-symmetric permutation to such a parabolic quotient (Section 5.1).In particular, it turns out that the parabolic quotients are determined by a type-B composition α of n, i.e., an integer composition of n with a possible additional 0-component.We denote such a parabolic quotient by H α .
Next, we study the longest element in H α and describe its c-sorting word together with the associated inversion order by means of skew shapes (Section 5.2).We then characterize the c-aligned elements associated with a parabolic quotient of the hyperoctahedral group algebraically in terms of a forcing condition on the inversions and combinatorially in terms of a pattern-avoidance condition (Section 5.3).It shall be remarked here that there are two types of type-B compositions: those with a 0-component and those without.This dichotomy has an impact on the pattern-avoidance condition, because the avoided patterns differ slightly depending on whether the 0-component is present or not.Nevertheless, we may write Align H α , w o;α ( c) uniformly for the set of c-aligned elements in H α .
In the last section of this article (Section 6), we prove that Align H α , w o;α ( c) induces a quotient lattice of the weak order on H α , called the type-B parabolic Tamari lattice Tam B (α).
Theorem 1.1.For every type-B composition α, Tam B (α) is a lattice.Moreover, it is a quotient lattice of the weak order on the parabolic quotient H α .
In fact, congruence-uniformity of Tam B (α) follows right away from the fact that Tam B (α) is a quotient lattice of the weak order because the latter is itself congruence uniform.Since trimness of Tam B (α) cannot be deduced so easily, we think that it is justified to state these important structural properties together as a theorem.
We conclude this article with a brief outlook on future research in this direction (Section 7).

POSETS AND LATTICES
2.1.Partially ordered sets.Throughout this section, let P denote a finite set.If P is equipped with a reflexive, antisymmetric and transitive relation ≤, then P = (P, ≤) is a partially ordered set (or poset for short).An induced subposet of P is a poset (Q, ≤ ′ ) for Q ⊆ P, where q ≤ ′ q ′ if and only if q ≤ q ′ for all q, q ′ ∈ Q.
The direct product of P with a poset (P ′ , ≤ ′ ) is the poset (P × P ′ , ) with (p 1 , p ′ 1 ) (p 2 , p ′ 2 ) if and only if p 1 ≤ p 2 and p ′ 1 ≤ ′ p ′ 2 .An element p ∈ P is minimal if for every q ∈ P with q ≤ p it follows that p = q.It is maximal if for every q ∈ P with p ≤ q it follows that p = q.If P has a unique minimal and a unique maximal element, then P is bounded.Then, we usually write 0 (resp.1) for the unique minimal (resp.maximal) element of P.
An interval of P is a set [p, q] def = {r ∈ P : p ≤ r ≤ q} for some p, q ∈ P with p ≤ q.A covering pair (or cover for short) of P is a pair (p, q) ∈ P × P such that p < q and there does not exist an r ∈ P such that p < r < q.We usually write p ⋖ q in this situation.In other words, a covering pair induces an interval of cardinality two.
A subset C ⊆ P is a chain if we can write C = {c 1 , c 2 , . . ., c k } such that c 1 < c 2 < • • • < c k .A chain is maximal if there exists no strictly larger chain C ′ with C C ′ .In other words, a maximal chain C can be written as C = {c 1 , c 2 , . . ., c k } with c 1 ⋖ c 2 ⋖ • • • ⋖ c k , where c 1 is minimal and c k is maximal.The length of P is ℓ(P) def = max |C| − 1 : C is a maximal chain of P .
If (P ′ , ≤ ′ ) is another poset, then a map f : P → P ′ is order preserving if for all p, q ∈ P with p ≤ q it holds that f (p) ≤ ′ f (q).

2.2.
Lattices.An element p ∈ P is a lower bound for a subset A ⊆ P if p ≤ a for all a ∈ A. The greatest lower bound of A is (if it exists) the unique maximal element of the set of lower bounds of A (considered as a subposet of P).Upper bounds and least upper bounds are defined analogously.
Then, P is a lattice if for all p, q ∈ P the set {p, q} has a greatest lower bound (the meet, denoted by p ∧ q) and a least upper bound (the join, denoted by p ∨ q).It is easy to see that every finite lattice is bounded.A sublattice of P is a subposet Q = (Q, ≤) which is itself a lattice and for every p, q ∈ Q the join (and meet) of p and q in Q is the same as the join (and meet) in P.
An element j ∈ P \ { 0} is join-irreducible if j = p ∨ q implies j ∈ {p, q} for all p, q ∈ P. Since P is finite, for every join-irreducible element j ∈ P there exists a unique element j * ∈ P such that j * ⋖ j.Meet-irreducible elements are defined analogously.Let J(P) and M(P) denote respectively the set of join-and meetirreducible elements of P.

2.3.
Congruence-uniform lattices.Let P = (P, ≤) be a lattice.A lattice congruence of P is an equivalence relation Θ on P that respects the lattice operations.More precisely, for p 1 , p 2 , q 1 , q 2 ∈ P with (p 1 , q 1 ) ∈ Θ and (p 2 , q 2 ) ∈ Θ, we have (p 1 ∨ p 2 , q 1 ∨ q 2 ) ∈ Θ and (p 1 ∧ p 2 , q 1 ∧ q 2 ) ∈ Θ.The set of congruence classes of Θ forms a lattice again, called the quotient lattice P/Θ.The order relation is given by comparing representatives of each congruence class.
The following characterization of lattice congruences will be useful later.
If f is surjective, then the preimages of f induce a lattice congruence on P.Moreover, given a lattice congruence Θ, the map that sends p ∈ P to the minimal element of [p] Θ is a surjective lattice homomorphism.
We may as well consider the set of all congruence relations on P ordered by refinement.This is again a lattice [12]; the congruence lattice of P, denoted by Con(P).From this perspective, a lattice congruence Θ of P is join-irreducible if it is a joinirreducible element of Con(P).For p ⋖ q, let cg(p, q) denote the finest lattice congruence on P in which p and q are congruent, i.e., the intersection of all lattice congruences in which p and q are congruent.Interestingly, the covering pairs of P determine the join-irreducible lattice congruences.Theorem 2.2 ([13, Section 2.14] and [11,Theorem 3.20]).Let P be a finite lattice and let Θ ∈ Con(P).The following are equivalent.
As a consequence, any finite lattice P admits a surjective map cg : J(P) → J Con(P) , j → cg(j * , j), which in general fails to be injective.A lattice P is congruence uniform if the map cg is a bijection for P and its dual P d = (P, ≥).

Proposition 2.3 ([25]
).The class of congruence-uniform lattices is preserved under lattice homomorphisms, under passing to sublattices, and under taking (finitely many) direct products.

Trim lattices.
According to [20], a lattice P is extremal if J(P) = ℓ(P) = M(P) .However, every lattice can be embedded as an interval into an extremal lattice; see [20,Theorem 14(ii)].Thus, extremality alone is not necessarily inherited by intervals.
A lattice property, which is stronger than extremality and behaves well with respect to sublattices and quotients, was introduced in [31].An element p ∈ P is left modular if for all r, q ∈ P with r ≤ q it holds that Then, P is left modular if it has a maximal chain consisting of ℓ(P) + 1 left-modular elements.Left-modular lattices have interesting topological properties, though not in the scope of this paper; see for instance [19,21].We are rather interested in lattices that are both extremal and left modular; following [31] we call such lattices trim.
Proposition 2.6 ([31, 32]).The class of trim lattices is preserved under lattice homomorphisms, under passing to intervals, and under taking (finitely many) direct products.
It is, however, not necessarily true that any sublattice of a trim lattice is trim again, see [31,Theorem 3].For semidistributive lattices it is somewhat easier to verify trimness, as we only need to establish extremality.Theorem 2.7 ([32, Theorem 1.4]).Every semidistributive and extremal lattice is trim.

BASICS ON COXETER GROUPS
3.1.Coxeter groups.We start with some algebraic background on Coxeter groups.A Coxeter group is a group W with identity element e and a distinguished set S = {s 1 , s 2 , . . ., s n } of (Coxeter) generators that admits a presentation of the following form: The parameters m i,j form the Coxeter matrix of W. Note that m i,j = ∞ is allowed if there is no relation between the generators s i and s j .
A word for w ∈ W is a representation of w by a group product w i 1 w i 2 • • • w i k , where w i j ∈ W for all j.Such a word is S-reduced if w i j ∈ S for all j and w cannot be expressed as a product of fewer than k generators.The number of letters in an S-reduced word for w is its Coxeter length, denoted by ℓ S (w).If W is finite, then there exists a unique element ω o , called the longest element, which maximizes ℓ S .
3.2.The weak order.The (left) weak order on W is defined by The weak order on W constitutes a graded poset whose rank function is precisely ℓ S .This order is called left weak order, because we have a covering pair u ⋖ weak v if and only if there exists s ∈ S such that v = su and ℓ S (v) = ℓ S (u) + 1.In other words, covering pairs are determined by left multiplication with a Coxeter generator.We sometimes write Weak(W) instead of (W, ≤ weak ).
Remark 3.1.We may as well define a right weak order on W in an analogous fashion.The two orders are isomorphic via reversal of S-reduced words.For our purposes, it will be more convenient to work with the left weak order, even though the right weak order is more frequently used in the literature.
For our purposes, the following two properties of the weak order are of major interest.

Reflections and inversions.
The weak order on W can be expressed equivalently via inclusion on special set systems.To explain this connection, we define the set of reflections of W by Clearly, S ⊆ T and we usually call the elements of S the simple reflections of W.
A (right) inversion of w ∈ W is a reflection t ∈ T such that ℓ S (wt) < ℓ S (w), and we define the (right) inversion set of w by We have the following correspondence between inversion sets and length of Sreduced words.Lemma 3.4 ([2, Corollary 1.4.5]).For w ∈ W, we have Inv(w) = ℓ S (w).
A cover inversion of w is t ∈ Inv(w) such that there exists s ∈ S with sw = wt.The name comes from the fact that, by definition, the cover inversions of w are in bijection with the elements covered by w in Weak(W).The cover inversion set of w is denoted by Cov(w).

Roots and aligned elements.
A Coxeter group admits a representation as a group of reflections acting on a Euclidean vector space [16]; these reflections correspond bijectively to the elements of T. Such a reflection sends a non-zero vector to its negative and fixes a hyperplane pointwise.We may choose two (mutually inverse) normal vectors to every such hyperplane and obtain the root system associated with W.Moreover, by choosing an appropriate half-space, we may partition the root system into positive and negative roots; the collection of positive roots is denoted by Φ + W .This sets up a bijection t → α t from T to Φ + W .Let us, conversely, denote by t α the reflection corresponding to the positive root α ∈ Φ + W .Let w = a 1 a 2 • • • a k be an S-reduced word for w ∈ W. For i ∈ [k], we define Inv(w) = {t 1 , t 2 , . . ., t k }.
In fact, we obtain a linear order on Inv(w) in this way, called the inversion order of w, denoted by Inv(w).Combinatorially, we may realize H n as follows.For n > 0 we define the set of signed integers by Then, equivalently, H n is the group of all sign-symmetric permutations of ±[n]; we usually call it the hyperoctahedral group of degree n.It is easily checked that Let [[i]] denote the sign-symmetric permutation that exchanges i and −i, and ((i j)) the sign-symmetric permutation that exchanges the values i and j (and simultaneously −i and −j), where we do not make assumptions about the sign of i and j.Then, the assignment s 0 → [ [1]] and s i → ((i i+1)) for i ∈ [n − 1] allows us to switch from the Coxeter representation of H n to its permutation representation (when multiplying by s i on the left of a sign-symmetric permutation).Via the induced isomorphism, we may transfer the notions of reflection set, Coxeter length, (cover) inversions, and weak order to sign-symmetric permutations.
Note 4.1.The set of reflections of H n can be partitioned into three types: the ones that exchange i with −i, the ones that exchange two positive values (and their opposite values), and the ones that exchange a positive and a negative value (and their opposite).They will respectively be denoted by [[i]], ((i j)), and ((−j i)).Since changing the signs of all elements or exchanging both elements represent the same reflection, we shall write the reflections in the following non-ambiguous way Note that, by definition, every reflection is itself an involution.
The length of π is as usual its number of inversions.Moreover, • Proof.For the definition of the (right) inversions, consider a sign-symmetric permutation π.Then one easily checks that multiplying it by a generator s i on its right either adds or subtracts one to its length.Moreover, if its length is not zero, there is always a way to subtract one to its length by applying a suitable generator, that is, if there is an inversion between i and j, then there exists an index k such that k and k + 1 is an inversion.Since composing π ∈ H n with [[i]] on the right effectively swaps the sign of π(i); composing π ∈ H n with ((i j)) effectively exchanges the entries of π in the i th and j th position, the first part of the claim holds.Now, concerning the cover inversions, one just has to decode the action of the s i on the left of a sign-symmetric permutation: s 0 exchanges the values 1 and −1, and s i exchanges the values i and i + 1 along with −i and −i − 1, hence the statement.• i = 0 and π(1) < 0; or • i = 0 and either π(i) > 0 and π(i) > π(i + 1), or π(i) < 0 and π(i) < π(i + 1).
We usually represent sign-symmetric permutations via their long one-line notation, i.e., for π ∈ H n we write down the values π(−n), π(−n+1), . . ., π(−1), π(1), . . ., π(n−1), π(n) in that order.We represent negative values by an overbar rather than a minus sign for stylistic reasons, and we add a vertical bar between π(−1) and π(1) to emphasize the symmetry.If no ambiguities may arise, we omit the commas in this sequence.The right part of π ∈ H n is the sequence π(1), π(2), . . ., π(n).Clearly, the right part determines π due to the sign-symmetry.We shall make use of the following notation: One easily finds the cover inversions of a permutation in this notation: start with 1 and then read 1, 2, and so on.If i + is to the left of i, their positions form a cover inversion.
One easily checks that π = s 0 s 1 s 2 s 7 s 5 s 3 s 2 s 0 .Then, see for instance [14, Section 5.3] for a detailed explanation.Let us consider the set Then, the subset of simple roots consists of linearly independent vectors and spans Φ + .In particular, expanding any α ∈ Φ + in terms of Π yields a linear combination with nonnegative coefficients.This is enough to conclude that Φ + is indeed a choice of positive roots for H n .We may equivalently see this by observing that if v = (1, 2, . . ., n) ∈ R n , then any α ∈ Φ + has (α, v) > 0 and any α ∈ Φ \ Φ + has (α, v) < 0 with respect to the standard scalar product of R n .
In the following lemma, we abbreviate α i,j Lemma 4.5.Let α ∈ Φ + , and let i, k ∈ [n] with i < k.The following list of decompositions of α as nonnegative linear combinations of two positive roots is exhaustive.
Proof.This is a straightforward computation.Note that for i = j, we get α i,j = 2ε i in the second decomposition of the third case.
In order to match Φ + with the reflections of H n , we identify a positive root ε i with the reflection [[i]], a positive root −ε i + ε j with the reflection ((i j)), and a positive root ε i + ε j with the reflection ((−j i)).

4.3.
A special family of aligned sign-symmetric permutations.We consider the The c-sorting word of w ∈ H n is the S-reduced word for w that appears as rightmost as possible in the half-infinite word consisting of infinitely many copies of the reverse of c.The c-sorting word of w is denoted by w( c).In order to describe the inversion order of w o ( c), we prove the following general lemma regarding c-sorting words of particular elements of H n .For i < k, let us abbreviate Lemma 4.6.If the right part of a sign-symmetric permutation w ∈ H n can be written as the concatenation of intervals in the following form then its c-sorting word is obtained by reading the following list from top to bottom, left to right: (3) Proof.Given that ∞ c is the repetition of the same sequence s n−1 • • • s 1 s 0 , the rightmost reduced word of w in it is obtained by computing, given a sign-symmetric permutation π and a value j, the first value k within the ordered sequence j, . . ., n − 1, 1, . . ., j − 1 such that ℓ S (πs k ) < ℓ S (π).The induction begins with the pair (w, 0).Now, the property holds by an easy induction on r: one gets again such an element by multiplying w on the right by the sequence of s i of the last line of (3) read from right to left.This product is the rightmost one in ∞ c since each product s 0...n−j first changes the sign of the first letter then moves it to the right until it is followed by a letter greater than itself or at the right extremity of the permutation.Since at the right of its final position we have an increasing sequence, the next possible generator we can use is s 0 , hence the induction.
Note that each product by an s k either changes the first value into a positive one or exchanges it with a letter smaller than itself so that the length of the element decreases at each step, hence showing that the final result is indeed a reduced word of w.
Starting from the c-sorting word of ω o ∈ H n we can explicitly compute the inversion order Inv w o ( c) .We defer a formal proof to Section 5.2.
For i ∈ [n], we consider the ordered list of pairs The n possible rows are now arranged such that first components increase from top to bottom and pairs with the same second component appear in the same column.
If we now read from top to bottom, right to left, then we obtain the inversion order Inv w o ( c) by identifying a pair (i, −i) with the reflection [[i]], a pair (i, j) for i > j > 0 with the transposition ((j i)) and a pair (i, −j) for i < j with the transposition ((−j i)).
Example 4.9.Let n = 4. Then c = s 0 s 1 s 2 s 3 and The previously mentioned arrangement of the sixteen possible pairs is: The replacement from above produces the following inversion order: We may thus characterize the members of Align H n , w o ( c) via their inversion sets.
Proof.Let t be a reflection of H n such that α t can be written as We use Lemma 4.5 and the description of Inv w o ( c) from above.To avoid lengthy conversions from reflections to roots, we use the notation "t 1 + t 2 " as a short-hand for "t α t 1 +α t 2 ".We consider the possible cases, and pick i, k ∈ then by Lemma 4.5, we have t = ((j i)) + [[j]] for 0 < j < i.In the inversion order, we clearly have , then by Lemma 4.5, we have t = ((i j)) + ((j k)) for i < j < k.In the inversion order, we clearly have , then by Lemma 4.5, we have the following three options. (a In the inversion order, we have In the inversion order, we have These considerations show that if t ∈ Cov(π), then Inv(π) must contain the specified inversions for π to be w o ( c)-aligned.
Using the permutation representation of H n , we can identify w o ( c)-aligned elements in terms of pattern avoidance.A permutation π ∈ H n has a type-B 231- pattern if there exist indices −n ≤ i < j < k ≤ n such that π(i) = π(k) + as well as π(j) > π(i) and j, k > 0. In other words, we look for subwords of the long oneline notation of π standardizing to 231 with the requirement that the positions of the '2' and the '1' form a cover inversion and and the positions of the '3' and the '1' are positive.Let H n (231) be the set of sign-symmetric permutations without a type-B 231-pattern.We get and Comparing this with Lemma 4.10, we note that π is w o ( c)-aligned, and that π does not have any type-B 231-pattern.There are three "usual" 231 patterns, determined by the positions (4, 3, 5), (1, 2, 2) and (1, 2, 1).However, in each case the middle position is negative, and therefore none of them is a type-B 231-pattern.
On the other hand, consider the sign-symmetric permutation We get Comparing this with Lemma 4.10, we observe that σ is not w o ( c)-aligned, because for instance

and only if π does not have a type-B 231-pattern.
Proof.This is immediate from Lemmas 4.2 and 4.10.
The weak order on the set H n (231) inherits some remarkable properties from the weak order on the full group H n .Let us call the poset the type-B Tamari lattice.
Figure 1 shows the lattice Weak H 3 where we have highlighted the intervals characterized by the projection map that sends an element to the largest aligned permutation smaller than itself.Figure 2 shows Tam B (3), which is the quotient lattice induced by the highlighted intervals in Figure 1.In these figures, we have colored the positions in the long one-line notation by symmetrically placed colors.This foreshadows our representation of the elements in parabolic quotients of H n in the next section.Theorem 4.13 ([15,26]).For n > 0, the poset Tam B (n) is both a sublattice and a quotient lattice (for the projection sending a sign-symmetric permutation to the largest aligned permutation smaller than itself) of Weak(H n ).Moreover, it is congruence uniform and trim.

PARABOLIC QUOTIENTS IN TYPE B
5.1.A distinguished set of representatives.Let W be a Coxeter group, whose set of simple reflections is S. Any J ⊆ S generates the parabolic subgroup W J of W. The parabolic quotient W J def = W/W J consists of the left cosets of W J in W. By construction, every such coset has a member of minimal Coxeter length.In the following, we identify the quotient W J with this set of minimal length representatives.Equivalently, we can define the set W J as follows: If we consider the Coxeter group of type B in its permutation representation defined above, we may describe the parabolic quotients neatly as follows.Let n > 0, and consider a composition α = (α 1 , α 2 , . . ., α r ) of n, i.e., A type-B composition of n > 0 is a composition of n with a possible zero-component α 0 def = 0 in the beginning.A type-B composition is split if it has a zero-component, and it is join otherwise.
The number of type-B compositions of n is 2 n , because any type-B composition α is associated with a unique subset J α ⊆ S. The non-zero-components of α determine a subset of {s 1 , s 2 , . . ., s n−1 } as before, and we add s 0 to this set if and only if α is split.
Therefore, type-B compositions are in natural bijection with the subsets of the simple reflections of H n , so that if α is a type-B composition, then there exists a unique parabolic quotient of H n determined by the complement S \ J α .We write H α for the resulting set of minimal-length representatives (see Example 5.3).
Remark 5.1.Let us emphasize the difference between the notions H n where the subscript is an integer, and H α where the subscript is a type-B composition.The first denotes the full hyperoctahedral group of degree n and the second denotes the set of minimal length representatives of the parabolic quotient associated with α.
Let α be a type-B composition with r non-zero-components α 1 , α 2 , . . ., α r appearing in that order.As before, we define if α is join.
Figure 1.The weak order on H 3 .
This means, if α is split, that we partition ±[n] into 2r symmetrically placed blocks, and if α is join, we partition it into 2r − 1 blocks, where the middle block contains positive and negative positions.This partition also explains the terminology "join" and "split" for the type-B compositions.The partition corresponding to a join composition has a block joining positive and negative positions, the partition corresponding to a split composition does not.
For an index a ∈ [n] of the right part, we say that a is in the i th α-region if p i−1 ≤ a < p i , where p 0 = 0. We sometimes write ̺ α (a) = i in this situation.

Lemma 5.2. Let α be a type-B composition. A sign-symmetric permutation π ∈ H n belongs to H α if and only if the long one-line notation of π, when partitioned according to
Part(α), is increasing in each block.
Proof.Let α be a type-B composition, and let J α be the corresponding subset of the simple reflections S = {s 0 , s 1 , . . ., s n−1 }.Note that we use the set S \ J α to construct the parabolic quotient H α .
Let π ∈ H n and s ∈ S be such that ℓ S (π ′ ) > ℓ S (π) for the product π ′ = πs.Since s is an involution, we get s ∈ Inv(π ′ ), and according to Lemma 4.2 we have π ′ (1) Comparing this with the definition of Part(α) yields the claim.
Example 5.3.Let n = 3 and consider α = (0, 1, 2).Then α is split and we obtain J α = {s 0 , s 1 }, which means that S \ J α = {s 2 }.The long one-line notation is therefore partitioned into four parts: {−3, −2}, {−1}, {1}, {2, 3}.We highlight the first and last part in green and the second and third part in orange.Then, H α has the following 24 elements: If we choose α ′ = (1, 2), then α ′ is join and we obtain J α ′ = {s 1 } which gives us S \ {J α ′ } = {s 0 , s 2 }.We may partition the long one-line notation into three parts {−3, −2}, {, −1, 1}, {2, 3}, and we highlight the first and the last part in green and the middle part in orange.The set H α ′ consists of the following twelve elements: Example 5.5.We continue Example 5.3 with α = (0, 1, 2), and observe that its It is our declared goal to describe the w o;α ( c)-aligned elements of H α , and we therefore need a good understanding of the parabolic longest element ω o;α (which is the element w J • from Theorem 5.4 for the parabolic quotient H α ) and its c-sorting word.The long one-line notation of ω o;α is obtained by filling the blocks of Part(α) from left to right with the largest available values so that the increasing property per block from Lemma 5.2 is satisfied.This is made precise in the following lemma.
Lemma 5.6.Let α be a type-B composition, and let ω o;α be the longest element of H α .For a ∈ [n] with ̺ α (a) = i, we have Proof.This is a straightforward computation.
For the description of the c-sorting word of ω o;α , it will be convenient to model it combinatorially using certain skew shapes.This is foreshadowed in the arrangement of the inversions of ω o for the full hyperoctahedral group H n after Corollary 4.8.
Given integers λ 1 ≥ λ 2 ≥ • • • , a Ferrers diagram of shape (λ 1 , λ 2 , . ..) is a leftaligned arrangement of unit boxes, where the i th row consists of λ i boxes.We use the English notation, i.e., the first row is at the top.We usually do not distinguish between the Ferrers diagram and the integer partition it represents.
For two Ferrers diagrams λ, µ, the associated skew diagram λ/µ consists of all boxes of λ that are not boxes of µ.See Figure 3 for an illustration.
In other words, the k th entry of µ is the sum of the sizes of the α-regions starting from the one containing k. Now, we consider the Ferrers diagram λ (α) = (λ 1 , λ 2 , . . ., λ n ), where We write skew(α) def = λ (α) /µ (α) .We now construct a filling of skew(α) as follows.Let k ∈ [n].If α is split or k > α 1 , then we fill the k th row of skew(α) with the Coxeter generators s 0 , s 1 , . . .from right to left until all the boxes in that row are exhausted.If α is join and k ≤ α 1 , then we fill the k th row of skew(α) with the Coxeter generators s α 1 −k+1 , s α 1 −k+2 , . . .also from right to left until all the boxes are exhausted.We denote by w α the word obtained by reading the Coxeter generators in skew(α) from bottom to top, left to right.
Proposition 5.8.For every type-B composition α, the word w α is the c-sorting word of ω o;α .
Proof.If α is split, we already are in the case of Lemma 4.6 with i 1 = 0.If α is join, then the first values on the right part of w are positive, and the first generator one can apply is at the end of the first α-region.Then this value goes to the extreme right of the permutation and the same idea applies again to the other elements of the first α-region: they move to the right until they get next to a value greater than themselves.This exactly accounts for the first i 1 rows of skew(α) read from right to left.Again, each product decreases the length of the permutation, hence proving that we have a reduced word.After that product of generators, we are back with a permutation of the type shown in Lemma 4.6, hence the result.Corollary 5.9.We shall need the following lemma whose proof is immediate by direct computation.
Lemma 5.12.Consider the product (6) s It sends k > 0 to For example, with n = 7 and i = 3, one gets s Proof of Proposition 5.11.Recall that each inversion is obtained by conjugating an s i and so is itself an reflection, and in particular an involution.And an involution that is not the identity is characterized by the image of an element not sent to itself.
Let us start with a split composition α = (0, α 1 , α 2 , . . ., α r ) of n and compute the inversion corresponding to an s i in the k-th row of part α j (so that k ≤ α j ).Note that i + k ≤ n since each row has one element less than the previous one inside s α j .
Let us denote by π the permutation defined by the product Then the involution we want to compute is w = π −1 s 0...i−1 s i s i−1...0 π.We are considering an element in row t = α 1 + α 2 + • • • + α j−1 + k (from the top).Let us compute w(t).First, π(t) = 1: each product s p...0 decreases the value by 1, and since there are t − 1 such products, we get 1.Then s 0..
We now need to evaluate the remaining rows of π −1 on −i − 2 which is equal to the image of t − 1 by the conjugate of the generator s i+1 in row k − 1 of α j , hence showing that the columns of the skew shape of α are filled with the same values within the same part of the composition.Now, if k = 1, there are again two cases.Either i + 1 ≤ n − α j−1 or not.If it is the case, that exactly means that the cell we consider has a cell above it and the same reasoning as before applies.Otherwise, we are in the topmost cell of its column.In that case, s 0...α j−1 (−i − 1) is either 1 1 to this result will eventually get the value 1 and then increment it at each product by an s 0...p .Moreover, applying then the other elements of π −1 , it will still increment on each row, finally giving the value n − which is exactly the value of its column in skew(α).
Let us now consider the join case.The labels of the rows of the first part go backwards, which is easily explained (see the proof of Proposition 5.8).Once that is noted, if the cell under consideration has a cell above it, the same argument as before works and one easily shows that the skew shape predicts the correct involution.If the cell is the topmost of its column, the explanation we saw before in the case of the split composition will apply again with one small difference: the first part behaves differently from the next ones since it is not multiplied on its left by a sequence s k but by a sequence that does not contain the generator s 0 .That explains why their images do not change sign, in contrast to other parts of α.

w o;α ( c)-aligned elements for parabolic quotients of H n .
In this section we want to generalize Lemma 4.10 to parabolic quotients of H n .So, fix a type-B composition α, and let π ∈ H α .Let t ∈ Cov(π).We use the inversion order Inv w o;α ( c) to describe the required implications.
In the following proofs we will simply write "aligned" when we mean "w o;α ( c)aligned".Moreover • ) and we may discard this decomposition.Otherwise [[j]] appears in a row above [[i]] and ((j i)) is to the left of [[i]] in the same row.Therefore, we have [[j]] ≺ t ≺ ((j i)), and for π to be aligned, we need to have . By Lemma 4.5, we have t = ((i j)) + ((j k)) for i < j < k.By construction, ((i j)) is in the row labeled −j and the column labeled by −i, while ((j k)) is in the row labeled −k and in the column labeled by −j.Now, j < k implies that ((i j)) lies above t in skew(α), while ((j k)) lies to the left of t.We thus get the order ((i j)) ≺ t ≺ ((j k)).So, for π to be aligned we need that ((i j)) ∈ Inv(π) so that we may discard the corresponding decomposition.Analogously, we may discard the case ̺ α (j) = ̺ α (k).
If we take t = [ , so for π to be aligned we need Once again, for π to be aligned we need ∈ Inv(ω o;α ), so that we may discard this decomposition.Otherwise, ((−j i)) appears in a row above ((j k)), because i < k.Indeed, if i < j, then ((−j i)) appears in the row labeled by −i, and if j < i then ((−j i)) appears in the row labeled by −j, which is above the row labeled −i.Therefore, we have ((−j i)) ≺ t ≺ ((j k)) and for π to be aligned, we need to have ((−j i)) ∈ Inv(π).
Lastly, say that t = (( ∈ Inv(ω o;α ) and we may discard this decomposition.If ̺ α (j) < ̺ α (i), then ((j i)) is in the row labeled −i and in the column labeled −j, and therefore it is to the left of t, which is in the row labeled −i and in the column labeled k.Since j < i < k, we have that ((−k j)) is in the row labeled −j, which is above the row containing t.Thus, we get ((−k j)) ≺ t ≺ ((j i)).For π to be aligned we need to have ((−k j)) ∈ Inv(π).Note 5.14.When a relation is implied by another, we also say that it is forced by that.Since the inversions are put in a skew partition by rows from the top-right corner cell to the left-bottom one, the alignment condition of Lemma 5.13 can be translated to: any inversion forced by a given inversion is either in the same row as the given inversion and to its right, or in the same column and above it, as one can check on Example 5.10.
Combinatorially speaking, we model aligned elements for H α with α split as follows.
Definition 5.15.Let α be a split type-B composition of n.A type-B (α, 231)-split pattern of a sign-symmetric permutation π Proof.According to Lemma 5.13, we only need to show that the violation of any condition is equivalent to the existence of a type-B (α, 231)-split pattern.We first consider the violation of conditions in Lemma 5.13.

The join case.
Let us now consider the join case.
Lemma 5.19.If α is join, then π ∈ H α is w o;α ( c)-aligned if and only if for every cover inversion t ∈ Cov(π) the following implications hold.
then we may discard this decomposition.If j ≤ α 1 , then by Lemma 5.18, we have ), and we may discard this decomposition as well.For α 1 < j < i, we see that [[j]] lies in the row labeled by −j, which is above the row labeled −i that contains [[i]].Moreover, ((j i)) lies in the row labeled −i, to the left of [[i]], which yields [[j]] ≺ t ≺ ((j i)).Thus, for π to be aligned, we must have [[j]] ∈ Inv(π).
As in the proof of Lemma 5.13, we may discard the cases where ̺ α (i) = ̺ α (j) or ̺ α (j) = ̺ α (k).Now, there are two options where ((i k)) can be.If i ≤ α 1 , then it is in the column labeled by k and in the row labeled by i.By construction, j > α 1 , such that ((i j)) lies to the right of ((i k)) in the same row, and ((j k)) lies in the row labeled −k and in the column labeled −j, which is below ((i k)).If i > α 1 , then ((i k)) lies in the row labeled −k and in the column labeled −i.Moreover, ((i j)) lies in the row labeled −j (which is above the row labeled −k) and ((j k)) lies in the row labeled −k and in the column labeled −j, to the left of ((i k)).Thus, in both cases we have ((i j)) ≺ t ≺ ((j k)), so that for π to be aligned, we need ((i j)) ∈ Inv(π).
2, so that we can discard this decomposition.Thus, ((−k i)) is in the row labeled −i.Now, ((j k)) is in the row labeled j, and thus above t.Since j ≤ α 1 , ((−j i)) is in the row labeled −i but to the left of t.Therefore, we have ((j k)) ≺ t ≺ ((−j i)).If j > α 1 , then ((−j i)) is in the column labeled by −j and ((j k)) is in the column labeled by −k.If i ≤ α 1 , then ((−k i)) is in the row labeled −k but to the right of ((j k)), and if i > α 1 , then ((−k i)) is in the row labeled −i, which is above the row labeled −j.Thus, we get ((−j i)) ≺ t ≺ ((j k)).Concluding, for π to be aligned we need ((j k)) ∈ Inv(π) when j ≤ α 1 , and ((−j i)) ∈ Inv(π) when j > α 1 .
Lastly, say that t = ((j i)) + ((−k j)) for 1 ≤ j < i.If ̺ α (i) = ̺ α (j), then we can discard this decomposition.Thus, ̺ α (j) < ̺ α (i).In any case, i > α 1 , which means that ((−k i)) is in the row labeled by −i.If j ≤ α 1 , then ((j i)) is in the row labeled j, and ((−k j)) is in the row labeled by −k.This yields ((j i)) ≺ t ≺ ((−k i)).If j > α 1 , then ((j i)) is in the row labeled −i but to the left of t, and ((−k j)) is in the row labeled by −j.Thus, we get ((−k j)) ≺ t ≺ ((j i)).Hence, for π to be aligned we need (( Note 5.20.As in the split case, when a relation is implied by another, we also say that it is forced by that one.Since the inversions are put in a skew partition by rows from the top-right corner cell to the left-bottom one, the alignment condition of Lemma 5.19 may be translated to: any inversion forced by a given inversion is either on the same row as the given inversion and to its right, or in the same column and above it, as one can check on Example 5.10.
Combinatorially speaking, we model aligned elements for H α when α is join as follows.
Definition 5.21.Let α be a join type-B composition of n.
Comparing with the definition of split patterns, we see that the only difference in the join case is that, when the middle index j is in the first region, the join one, π(j) must be smaller than the two consecutive values π(i), π(k) instead of being larger.In this situation, we actually see a 312-pattern.

Proposition 5.22. Let α be a join type-B composition of n. A sign-symmetric permutation π ∈ H α is w o;α ( c)-aligned if and only if it does not have a type-B (α, 231)-join pattern.
Proof.Comparing Lemma 5.13 and Lemma 5.19 and using the same arguments in the proof of Proposition 5.16 and the fact that π ∈ H α , we observe that π has a type-B (α, 231)-join pattern (i, j, k) with i < 0 and j > α 1 if and only if π violates one of the conditions in Lemma 5.19, except for the following.
• There are some 0 . We now only need to show that the violation of the conditions above is equivalent to the existence of a type-B (α, 231)-join pattern (i, j, k) with i < 0 and 0 < j ≤ α 1 .
Now assume that we have in π a type-B (α, 231)-join pattern (i, j, k) with i < 0 and 0 < j ≤ α 1 .By definition, we have If 0 < −i ≤ k, then π violating the first condition above with i ′ = −i, j ′ = j and k ′ = k.Note that in this case we cannot have j = −i, as then i, −i would be in the first region, and we would have π(k) < π(i) < π(j), which is impossible.
If −i > k, then it is the second condition that is violated with i ′ = k, j ′ = j and k ′ = −i.We thus have the equivalence.
Example 5.23.We now illustrate Proposition 5.22 using the composition α ′ = (4, 2, 2).The corresponding inversion order is shown in Example 5.10.It remains to illustrate examples of (α ′ , 231)-join patterns, where the middle index is in the first αregion.The other cases are analogous to the split case.We consider the following permutations of H α ′ : • We have ((−8 5)) ∈ Cov(π 1 ) and ((−8 1)) ∈ Inv(π 1 ), but (( Example 5.24.Let us come back to the subtlety mentioned in Remark 3.7, namely that for the definition of w-aligned elements, where w is the c-sorting word of the longest element in a parabolic quotient of W, only the cover inversions must pass a certain check.Let us start with the observation that, in the case where the parabolic quotient is the whole group (i.e., J = ∅), this difference is irrelevant because the conditions are equivalent, see [23,Section 6.2].
It can be verified that the w o;α ( c)-aligned permutation 3 1 2 2 1 3 exhibits the same behavior, so that only fourteen of the sixteen w o;α ( c)-aligned elements pass the more restrictive condition of checking all inversions in Definition 3.6.
Remark 5.25.In light of Example 5.24, we notice that changing the condition from Definition 3.6 to checking all inversions would in general produce a smaller set of "aligned" elements for parabolic quotients of a Coxeter group W. The reason why it is proposed in [23] to only verify the defining conditions for cover inversions lies in the numerical coincidence mentioned in [23,Conjecture 41], namely that we can construct other sets of "Coxeter-Catalan"-like objects which share the cardinality of Align W J , w J o (c) but not that of the smaller collection.(α, 231)-join) pattern when α is split (resp.join).We denote by H α (231) the set of such sign-symmetric permutations.By Propositions 5.16 and 5.22, the set H α (231) contains exactly the w o;α ( c)-aligned elements in H α .This fact leads naturally to the following definition.To show that Tam B (α) is indeed a lattice, we shall construct a surjective map from H α to H α (231) which we then use to prove the lattice property.We need the following lemma in our type-B case, which is similar to Lemma 12 in [23] for the type-A case.Lemma 6.2.For every π ∈ H α , there is a unique element π ↓ ∈ H α (231) with Inv(π ↓ ) ⊆ Inv(π) such that Inv(π ↓ ) is maximal by inclusion for any such element.
We first deal with the case where α is split.By Proposition 5.16, we have a type-B (α, 231)-split pattern (i, j, k) in π .Without loss of generality, we may assume that j is the smallest such index in its region.We have ̺ α (i) < ̺ α (j) < ̺ α (k).We know that ((i k)) / ∈ Cov(π ′ ), as the absent inversion that makes ((i k)) violating the conditions in Lemma 5.13 is still absent in π ′ .We thus have π ′ (i) Suppose that our claim holds for some 0 , we have k d+1 < j, which is impossible due to the minimality of j.We thus have ̺ α (k d+1 ) > ̺ α (j) to conclude the induction.But this is absurd, as it implies ̺ α (i) When α is join, the only difference is when ̺ α (j) = 1 in the type-B (α, 231)-join pattern (i, j, k).In this case, we take j the maximal such index in the same region, and now we have π ′ (j) We observe that all patterns in the proof for α split above involving j remain valid, except that for the case ̺ α (k d+1 ) = ̺ α (j).In this case, as violating the maximality of j here.The modified argument thus applies to α that is join.We thus conclude that such π ′ with t ∈ Inv(π ′ ) cannot exist, whether α is split or join.This concludes the induction.Remark 6.3.We saw in the proof of Lemma 6.2 that, to construct π ↓ from π, we only need to spot some type-B (α, 231)-split (or join, depending on α) pattern (i, j, k), apply the inversion that exchanges π(i) and π(k), and then repeat the same procedure until there is no longer any such pattern.The result is π ↓ .Lemma 6.2 also ensures that all choices of inversion lead to the same π ↓ .By Lemma 6.2, we see π → π ↓ as a surjective map from H α to H α (231).Definition 6.4.We define the downward projection map being maximal, whose existence is ensured by Lemma 6.2.
We have the following immediate consequence of the proof of Lemma 6.2.
We denote by H α (312) the subset of elements in H α that avoid type-B (α, 312)-split or -join patterns, depending on α being split or join.
Similar to the case of type-B (α, 231) patterns, we have the following lemma, whose proof is essentially identical to that of Lemma 6.2.The only difference is that we are now looking at cover inversions with a smaller element in the middle, instead of a greater one in Lemma 6.2 (except in the case when 0 < j ≤ α 1 , where it is reversed).Lemma 6.10.For every π ∈ H α , there is a unique element π † ∈ H α (312) with Inv(π † ) ⊆ Inv(π) such that Inv(π † ) is maximal by inclusion for any such element.
For the comparable case, we may suppose that σ < weak π.The case σ ⋖ weak π follows from Proposition 6.14, and other cases follows from sucessive application of the cover case.
For the incomparable case, we observe that is the unique maximal element in H α (231) below both π and σ.With the same argument, but passing by ι and Lemma 6.10, we also have Figure 6 shows Tam B (1, 2) and Figure 7 shows Tam B (0, 1, 2) .We have therefore just proved Theorem 1.1.
An immediate consequence is the following property of Tam B (α).
More precisely, the sequence C of suffixes of the c-sorting word of ω o;α forms a maximal chain of Tam B (α). Proof.We shall show the more precise statement since it implies the first one.First, since Tam B (α) is a subposet of all type B permutations, its length cannot exceed the number of inversions of its maximal element ω o;α .So the considered chain C is of maximal available length and it is a chain since the whole c-sorting word is a reduced decomposition so that each new letter exactly adds one inversion.Now, all the suffixes of C are aligned.Indeed, let π be such an element.Its inversions form a suffix of the list of inversions of ω o;α , or, in terms of the skew partition containing the inversions, they satisfy that if a cell c is a inversion of π, then any cell above of c or in its row to its right is also an inversion of π.But, thanks to Lemmas 5.13 and 5.19, a permutation is aligned if and only if for any t ∈ Cov(π), it implies other inversions in π and thanks to Notes 5.14 and 5.20, π satisfies all forcing relations.

Proposition 6.20. For every type-B composition α of n, the number of join-irreducible elements of Tam
The proof of this proposition directly comes from the next lemma.Indeed, recall that join-irreducible elements have by definition exactly one cover inversion.Since a cover inversion is in particular an inversion, the set of possible cover inversions of the join-irreducibles of Tam B (α) is the set of inversions of ω o;α .The next lemma shows that, given any such inversion ((i j)), there is exactly one join-irreducible element of Tam B (α) such that ((i j)) is its cover inversion, hence showing that the number of join-irreducible is equal to the length of ω o;α .Lemma 6.21.Let α be a type-B composition and let ((i j)) be an inversion of ω o;α .Then there is exactly one join-irreducible element π(i, j) of Tam B (α) such that ((i j)) is its unique cover inversion.
Moreover, it can be computed directly as follows.
• If i and j are both positive, fill from left to right the right part of a sign-symmetric permutation with letters starting from 1 in increasing order, forgetting about position i and the right of its block and ending at position j.Then fill the remaining positions from left to right with the remaining positive values in increasing order.• If i = −j, fill π with the values from 1 to n in that order the following positions: all the left part of π starting at i (or, if α is join, all the left part except the central block and then the right half of the central block instead) and then the right part of π starting at position j + 1. • If either i or j is negative and |i| < |j|, there are three different cases.
-If α is join and |i| > α 1 , then i is again the negative one.Put k = i + j and ℓ = j − α 1 .Then its right part is given by the sequence Proof.Let ((i j)) be an inversion of ω o;α and let us consider π a join-irreducible element with ((i j)) as its unique cover inversion.Let k and k + 1 be the values of this cover inversion, that is, the values in positions i and j.Since there is only one cover inversion, that means that the values 1 up to k are in increasing order from left to right in π and so are the values k + 1 up to n.
We shall distinguish cases depending on the values i and j, and show that π is unique.
First, consider the case i > 0 and j > 0. Then the positions of 1 and 1 do not form a cover inversion, so 1 is in the right part of π.Thus, the other positive values must also be in the right part of π: 1 up to k have to be on the right part, and since k + 1 is also on the right, all the other values are too.So any type-B 231 pattern can only occur with three values on the right half.We are thus in the type-A case and π is unique (see [23]).
Let us nonetheless prove by ourselves that π is unique, since we shall prove uniqueness by building it explicitly using ideas that will be used in other cases of this proof.Since there is only one cover inversion between values k and k + 1, all values to the left of k + 1 must be at most k − 1 and all values to the right of k must be at least k + 2.Moreover, all positions between the positions of k + 1 and k are filled with elements that are either smaller or greater than k, but not both: the values to the right of k + 1 in its block must be filled with greater elements whereas all others must be filled with smaller elements.In other words, given α and ((i j)), all positions are filled either with values greater or smaller than k, hence showing that the filling is unique and in particular that k is fixed.One easily checks that the element described in the statement is the correct one.Now, consider the case i = −j.Since it is a cover inversion, it has to be that π(j) = 1 and π(i) = 1.Since there are no other cover inversions, all positive values are in increasing order from left to right.
• If α is split, the values to the left of 1 in the right part of π must be negative.
Otherwise, we would have a type-B (α, 231)-pattern.So there are exactly n − 1 available positions to the right of 1 for the n − 1 positive values from 2 to n.So π is unique and it has no type-B (α, 231)-pattern since the right part of π is increasing.Moreover, it corresponds to the one in the statement.• If α is join, the same holds except for the central one; but in that case, only the right half of this part can be filled with positive values, again implying that we have to put n values in n given spots, hence only one solution.Again, it avoids any type-B (α, 231)-pattern (which is a 312 if the '1' is in the first block right half).
Let us finally consider the case where i and j have different sign and |i| < |j|.Let us first solve the case α split.In that case, i is the negative one; otherwise, we would have a type-B (α, 231)-pattern k . . .k . . .k+1.As before, since ((i j)) is the only cover inversion, 1 has to be in the right part of π, and so do all positive values up to k.Let now ℓ be the maximal value of the left part of π.Then the right part consists of three possibly intertwined sequences: ℓ . . .k+1, 1 . . .k, and ℓ+1 . . .n.But ℓ + 1 must be to the right of k, or we would have a type-B (α, 231)pattern k+1 . . .ℓ+1 . . .k; and 1 must be to the right of k + 1, otherwise we would have a type-B (α, 231)-pattern as k . . . 1 . . .k+1.So the three sequences are not intertwined at all: they have to be concatenated.But since the positions of k and k + 1 are given, one easily gets that ℓ = j and k = ℓ + i, so that π is unique and is the one described in the statement.
The join case when |i| > α 1 is essentially the same as in the split case up to one small difference: the first α 1 values of the first half have to be positive and greater than ℓ in order to avoid the type-B (α, 231)-pattern (which is actually 312).The other positions are filled as in the split case and one finds the sequence of the statement.
Finally, the join case when |i| < α 1 is a little different.First, in that case i is the positive one.The values 1 up to k must still be in the right part and in that order, so that they form a prefix of the right part (so that k = i).Now, there still remains to intertwine the sequences ℓ . . .k+1 and ℓ+1 . . .n.But again, there is no choice: the block containing α 1 must be filled after k by values greater than it, and the remaining part of the permutation must be increasing in order to avoid type-B (α, 231)-patterns, hence the solution of the statement.Proposition 6.23.For every type-B composition α of n, the lattice Tam B (α) is trim.
Proof.By Proposition 6.18, Tam B (α) is congruence uniform, and thus by Lemma 2.4 it is semidistributive.Now, Lemma 2.5 states that in Tam B (α) the number of joinirreducible elements agrees with the number of meet-irreducible elements.Propositions 6.19 and 6.20 thus imply that Tam B (α) is extremal.Using Theorem 2.7, we conclude that it is in fact trim.
We are now ready to conclude our second main result.
Proof of Theorem 1.2.Proposition 6.18 states that Tam B (α) is congruence uniform, which implies together with Lemma 2.4 that it is semidistributive.Proposition 6.23 states that Tam B (α) is trim.

CONCLUSION AND OUTLOOK
The constructions and results presented in this article are a natural follow-up to [23], where combinatorial realizations for parabolic aligned elements in linear type A were considered.Drawing inspiration from other results concerning the combinatorics of parabolic Catalan objects in linear type A there are some natural research problems that may be studied subsequently, see [6,10,22].
The first natural question to investigate is the exhibition of other combinatorial families associated with parabolic quotients of H n .In [6], tree and lattice path models for elements of parabolic quotients of the symmetric group were introduced, and these constructions can be extended to produce the corresponding models in linear type B. This is work in progress and will be addressed in a future paper.
Following the approach from [22] in linear type A, it will certainly be fruitful to study the secondary structures associated with the type-B parabolic Tamari lattices, such as their Galois graph, canonical join complex or core label order.This will also help in the research for a noncrossing partition model for parabolic quotients in linear type B.
One of the main results of [6] relates the parabolic Tamari lattices in type A to the ν-Tamari lattices of Préville-Ratelle and Viennot [24], see also [10].The underlying geometric structure was generalized to type B in [7], and it is a natural question to study if there is a similar connection with our type-B parabolic Tamari lattice.More precisely, we may study the following research question.Question 7.1.Let α be a type-B composition of an integer n.Is there some integer N such that the associated type-B parabolic Tamari lattice Tam B (α) is isomorphic to an interval in the type-B ordinary Tamari lattice Tam B (N)? Is it even possible that Tam B (N) contains all Tam B (α ′ ) as disjoint intervals, where α ′ ranges over all type-B compositions of n?
Besides the structural considerations, it may also be fruitful to consider various enumerative aspects of type-B parabolic Tamari lattices.For instance, the total sum of the cardinalities of parabolic aligned elements with respect to all parabolic quotients of a fixed symmetric group yields the sequence [30, A151498] and enumerates the dimensions of a certain family of Hopf algebras, see [1,6].The counterpart in type B is the sequence t 1 , t 2 , . .., where The first six terms of this sequence are 3, 15, 91, 598, 4109, 29071, which currently is not listed in the OEIS.Our work in progress on other combinatorial models for type B may give a possible explanation of these numbers.
A second intriguing enumerative problem is the determination of the cover enumerator of the type-B parabolic Tamari lattices.More precisely, if α is a type-B composition of n, then we define Consequently, H α (231) = ( 2n n−t ).In type A, the cover enumerator has been determined for certain parabolic quotients of the symmetric group in [18, Theorem 3.1] using a slightly different perspective.
We wish to end by observing that computational evidence leads us to suggest that the cardinality of H α (231), where α = (0, 1, 1, . . ., 1, 2) is a type-B composition of n is given by the type-D Catalan number 3n−2 n ( 2n−2 n−1 ).
Acknowledgements.HM has received funding from the European Research Council (Grant Agreement no.681988, CSP-Infinity).JCN was partially supported by the CARPLO project of the Agence Nationale de la recherche (ANR-20-CE40-0007).

Corollary 4 . 8 .
The c-sorting word of the longest element ω o ∈ H n is given by w
and we set p i def = −p i .We define Part(α) as the following partition of ±[n]:

Proof.Proposition 5 . 11 .
By definition, ℓ S (ω o;α ) is equal to the number of letters in the c-sorting word w o;α ( c), and by Proposition 5.8 this number is equal to the number of boxes in skew(α).Then the statement is immediate since we start with a skew arrangement of n rows, each consisting of n boxes, and then we remove a triangular part for each component of α on the left.If α is join, we remove another triangular part on the right.(i) If r > 0 and c > 0, then we fill this cell by ((r c)).This can only happen if α is join, and then necessarily r < c. (ii) If r < 0 and c = −r, then we fill this cell by [[r]].(iii) If r < 0 and c > 0 with c > −r, then we fill this cell by ((−c −r)).(iv) If r < 0 and c > 0 with c < −r, then we fill this cell by ((r c)).(v) If r < 0 and c < 0, then we fill this cell by ((−c −r)).By construction, we are in the first n columns, thus r < c.The filling of skew(α), read from top to bottom and from right to left, yields the inversion order Inv w o;α ( c) .

Figure 4 .
Figure 4.The weak order on H (1,2) with the congruence classes with respect to Θ highlighted.

Figure 5 .
Figure 5.The weak order on H (0,1,2) with the congruence classes with respect to Θ highlighted.

Inv(w)" in
Definition 43]).An element x ∈ W is w-aligned if x ≤ weak w and whenever t α ≺ t aα+bβ ≺ t β with respect to Inv(w) for a, b ∈ N, then t aα+bβ ∈ Cov(x) implies t α ∈ Inv(x).that the condition "t α ≺ t aα+bβ ≺ t β with respect to Definition 3.6 requires that {t α , t aα+bβ , t β } ⊆ Inv(w).We denote the set of w-aligned elements of W by Align(W, w).We shall see examples of aligned elements when W = B n (seeExample 4.11).

A permutation representation. The finite Coxeter groups were classified by Coxeter [8], and we are mainly concerned with the Coxeter groups of type B. If we consider the set S = {s 0 , s 1 , . . . , s n−1 }, then the group H n def = S : s 2
i = e, (s 0 s 1 ) 4 = e, (s i , s i+1 ) 3 = e for i > 0, (s i s j ) 2 = e for j > i + 1 .is the Coxeter group of type B.

Roots for the Coxeter group of type
B. For n > 0, and i ∈ [n], we denote by ε i the i th unit vector in R n .Then, it is well-known that the following is indeed a root system for the Coxeter group of type B: , most of the time Inv(ω o;α ) is a proper subset of the set of all reflections of H n .Therefore, if we have a reflection t bβ+cγ ∈ Inv(ω o;α ), where t β / ∈ Inv(ω o;α ) or t γ / ∈ Inv(ω o;α ), we do not need to consider the relative order of {t β , t bβ+cγ , t γ } ∩ Inv(ω o;α ) with respect to Inv w o;α ( c) .We then say that we discard this decomposition.If α is split, then π ∈ H α is w o;α ( c)-aligned if and only if for every cover inversion t ∈ Cov(π) the following implications hold.
1 by Lemma 4.2.By Lemma 5.18, we have i > α 1 , and by Lemma 4.5, we have t 2. By Lemma 4.5, there are four decompositions to consider.If we take t = [[i]] + [[k]], then Lemma 5.18 allows us to discard the case i ≤ α 1 .If i > α 1 , then by construction [[i]] lies above [[k]], and [[i]] lies in the same row as ((−k i)) but to the right.Thus we have [[i]] ≺ t ≺ [[k]], and for π to be aligned, we need to have It is clear that Θ is an equivalence relation, with Π α ↓ giving the smallest element of each equivalence class.We first show that each equivalence class of Θ is order-convex.For any π, σ, π ′ in H α with π The equivalence thus also holds.The equivalence relation Θ defined in Definition 6.7 is a lattice congruence on Weak(H α ), which is the interval [e, ω o;α ] in Weak(H n ), and the related quotient lattice is Tam B (α).To show that Θ is a congruence on Weak(H α ), we check that the conditions in Lemma 2.1 are satisfied.For each equivalence class of Θ, Lemma 6.2 implies the existence of a minimal element, which is in H α (231).Then Proposition 6.15 along with Lemma 6.10 shows the existence of a maximal element in each equivalence class.Along with Proposition 6.8, this concludes that equivalence classes of Θ are intervals, which is precisely the condition (i) in Lemma 2.1.The conditions (ii) and (iii) of Lemma 2.1 are ensured by Proposition 6.13.Taking the minimal element in each equivalence class, the quotient lattice is Weak(H α (231)) = Tam B (α). Proof.
By Theorem 3.3, Weak(H n ) is congruence uniform, and by Theorem 5.4, Weak(H α ) is an interval of Weak(H n ).Since intervals are sublattices, Proposition 2.3 implies that Weak(H α ) is congruence uniform, too.Proposition 6.16 now states that Tam B (α) is a quotient lattice of Weak(H α ), so that the claim follows by applying Proposition 2.3 once more.6.3.Extremality of Tam B (α).In this section, we prove that Tam B (α) is an extremal lattice, i.e., it has the same number of join-and meet-irreducible elements and this number agrees with its length.For every type-B composition α of n, the length of the lattice Tam B Proposition 6.18.For every type-B composition α of n, the lattice Tam B (α) is congruence uniform.Proof.