The Rank Enumeration of Certain Parabolic Non-Crossing Partitions

We consider $m$-divisible non-crossing partitions of $\{1,2,\ldots,mn\}$ with the property that for some $t\leq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapoton's $M$-triangle in this setting and conjecture a combinatorial interpretation for the $H$-triangle. This conjecture is proved for $m=1$.


Introduction
Non-crossing partitions have appeared in the combinatorial landscape in the early 1970s, and-despite their simple definition-have ever since served an important, connecting purpose for many different aspects of algebraic combinatorics. Even though they were considered under different names before (e.g., as planar rhyme schemes [6]), their systematic study has begun only after Kreweras' seminal article [26].
A (set) partition is a covering of the set [n] def = {1, 2, . . . , n} by non-empty, mutually disjoint sets (so-called blocks) and it is non-crossing if there do not exist indices 1 ≤ i < j < k < l ≤ n such that i, k belong to one block and j, l to another. We obtain additional structure if we order non-crossing partitions by refinement, i.e., one partition refines another if every block of the first partition is contained in some block of the second. In fact, the resulting partial order (denoted by ≤ ref ) endows the set of noncrossing partitions with a rank function (where the rank is n minus the number of blocks) and a lattice structure. This lattice of non-crossing partitions was studied intensively from enumerative and structural points of view in [14][15][16]34]. The notion of m-divisible non-crossing partitions goes back to Edelman [14], and refers to non-crossing partitions in which every block has size divisible by m.
A remarkable observation by Biane [9] (and independently Brady [10]) relates the lattice of non-crossing partitions to a certain interval in the Cayley graph of the symmetric group with respect to the generating set of all transpositions. This construction was generalised by Bessis [8] and by Brady and Watt [11] to finite irreducible Coxeter groups, for which combinatorial interpretations were given by Athanasiadis and Reiner [4,32]. Armstrong subsequently finished this stream of generalisation by detailing how m-divisible non-crossing partitions are constructed for finite irreducible Coxeter groups [1].
Most recently, Williams introduced a generalisation of non-crossing partitions to parabolic quotients of finite irreducible Coxeter groups [39], and this construction was studied later for parabolic quotients of the symmetric group in [30,31]. In this article, we study a certain family of these parabolic non-crossing partitions, namely those where one takes a parabolic quotient of the symmetric group with respect to an initial segment of the list of adjacent transpositions. This particular family of parabolic non-crossing partitions admits the following combinatorial interpretation. We fix positive integers n, t such that t ≤ n, and define a t-partition to be a partition of [n] in which no block intersects the set [t] in more than one element. Such a t-partition is non-crossing if there exist no four indices 1 ≤ i < j < k < l ≤ n such that either j ≤ t and i, l belong to one block and j, k to another, or t < j and i, k belong to one block and j, l to another. For t = 1 this construction reduces to the ordinary non-crossing partitions introduced in the first two paragraphs.
Our main result provides a closed formula for the number of multi-chains of mdivisible non-crossing t-partitions under refinement with prescribed ranks. For convenience, we denote the set of all m-divisible non-crossing t-partitions of [mn] by NC (m) n;t . Theorem 1.1. Let m, n, t, l be positive integers, and let s 1 , s 2 , . . . , s l+1 be non-negative integers with s 1 + s 2 + · · · + s l+1 = n − t. The number of multi-chains π 1 ≤ ref π 2 ≤ ref · · · ≤ ref π l , where π i ∈ NC (m) n;t and rk(π i ) = s 1 + s 2 + · · · + s i for i ∈ [l], is given by t(mn − t + 1) − s l+1 (t − 1) n(mn − t + 1) n s 1 mn s 2 · · · mn s l mn − t + 1 s l+1 . (1.1) The proof of Theorem 1.1 utilises generating functions and is carried out in Section 3, after the necessary definitions have been recalled in Section 2.
As an application of Theorem 1.1 we compute a certain bivariate integer polynomial, which can be seen as the generating function of the intervals of non-crossing m-divisible t-partitions under refinement, weighted by the Möbius function. We call this polynomial the M-triangle, denoted by M (m) n;t . In the case t = 1, it was observed by Chapoton (for m = 1) and Armstrong (for m > 1) that under certain substitutions of the variables, one obtains two other bivariate integer polynomials, called the F -and H-triangle, respectively [1,12,13]. These references also provide concrete combinatorial realisations of those polynomials.
We show in Section 4 that the same substitutions-when applied to M (m) n;t -yield integer polynomials, too, see Theorem 4.3. In the last part of this article, Section 5, we conjecture a combinatorial realisation of the H-triangle in terms of multi-chains of filters in a certain partially ordered set on transpositions, see Conjecture 5.6. We conclude by proving this realisation in the case m = 1 (again with the help of generating functions) using a lattice path model, see Theorem 5.13.

Preliminaries
2.1. Non-crossing partitions. Given an integer n > 0, a (set) partition is a family of non-empty, mutually disjoint sets, called blocks, whose union is all of [n] def = {1, 2, . . . , n}. For m > 0, a partition is m-divisible if every block has size divisible by m. Given a partition π, we write bl(π) for the number of blocks of π, and define the rank of π by   rk(π) def = n − bl(π). A partition π refines a partition π ′ if every block of π is contained in some block of π ′ ; we denote this relation by π ≤ ref π ′ .
A partition is non-crossing if there are no indices i < j < k < l such that i, k belong to one block and j, l belong to a different block. We denote the set of all m-divisible non-crossing partitions of [mn] by NC (m) n . Non-crossing partitions were introduced in [26], and the m-divisible case was studied in [14]. An excellent survey on m-divisible non-crossing partitions and various related objects is [1].
For t ∈ [n], a t-partition is a partition where no block intersects the set [t] in more than one element. A t-partition is non-crossing if there do not exist indices i < j < k < l such that either j ≤ t and i, l belong to one block and j, k belong to a different block, or j > t and i, k belong to one block and j, l belong to another block. We denote the set of all m-divisible non-crossing t-partitions of [mn] by NC (m) n;t . Clearly, we have NC (m) n . The non-crossing condition can be visualised in terms of arc diagrams. Given a tpartition π of [n], we draw n nodes, labelled by 1, 2, . . . , n, on a horizontal line, where we colour the nodes in [t] in white and the remaining nodes in black. Then, we connect two nodes i and j by an arc if and only if i and j are in the same block of π and every k ∈ {i+1, i+2, . . . , j−1} is in a different block. This arc stays below the nodes i+1, i+2, . . . , t and rises above the nodes max{i, t}+1, max{i, t}+2, . . . , j. (If t = 1, then we use the convention that all nodes are black.) See Figure 1. Then, an mdivisible t-partition of [mn] is non-crossing if and only if its arc diagram can be drawn in such a way that no two arcs cross.
In this article, we want to study the poset 1 of m-divisible non-crossing t-partitions under refinement. The next lemma enables us to reuse many of the well-known properties of the case t = 1. Proof. If π is a t-partition, then we define the transformed partitionπ via the map Essentially,π is obtained by rotating all the white nodes by 180 degrees counterclockwise. It is then clear that π ∈ NC    Proof. The claim that NC (m) n;t , ≤ ref is graded, and the claim that the rank of its elements is n minus the number of blocks follows from Lemma 2.1 and [14,Section 4]. It remains to determine the maximal rank of elements of NC (m) n;t , or equivalently the minimal number of blocks. But this is clearly t, since by definition each element of [t] must be in its own block, and this value is attained for instance by the partition {1, t+1, t+2, . . . , t+m−1}, {2, t+m, t+m+1, . . . , t+2m−2}, . . . , {t−1, (t−2)m+3, (t−2)m+4, . . . , (t−1)m+1}, {t, (t−1)m+2, (t−1)m+3, . . . , mn} .
Remark 2.3. For m = 1, we recover a particular case of the parabolic non-crossing partitions first considered in [31,39], and later studied in [30].
Remark 2.4. We may view the elements of NC (m) n;t as certain non-crossing partitions on an annulus. To that end, we place t nodes, labelled by 1, 2, . . . , t, on the inner boundary of an annulus (in clockwise order), and mn − t nodes, labelled by t+1, t+2, . . . , mn, on the outer boundary (in clockwise order, too).
Given an m-divisible partition π, we inscribe the convex hulls of the blocks of π on this annulus. If none of these convex hulls touches the inner boundary more than once, then π is in fact a t-partition. Modelling the non-crossing condition, however, is a bit more intricate. One requirement, certainly, is that no two convex hulls intersect in their interior. The other requirement is the following: if B is a non-singleton block containing the node i for i ∈ [t], let j be the smallest node in B \ {i}. This block casts a shadow on the nodes t+1, t+2, . . . , j−1 which is impenetrable for blocks containing i ′ for i < i ′ ≤ t. Indeed, if we suppose that there is a block B ′ containing i ′ and j ′ ∈ {t+1, t+2, . . . , j−1}, then the blocks B and B ′ would violate the non-crossing condition for t-partitions.
A t-partition is non-crossing if and only if it can be inscribed on an annulus in the previously described manner. See Figure 2a for an illustration of this construction for   can be inscribed on this annulus in a non-crossing fashion, too, see Figure 2b. However, it violates the shadow-condition: the block containing 1 and 6 casts a shadow on the nodes 4 and 5 which makes them invisible to node 3, while 3, 4 and 5 are in the same block. Indeed, if we unfold the annular diagram to an arc diagram, then the arc sequences representing the block {1, 6, 7, 8} and {3, 4, 5, 14} would cross. See Figure 1b. For work on non-crossing partitions on an annulus without the shadow-condition see [21] and the references contained therein.

2.2.
Weighted generating functions and Lagrange Inversion. Let x 1 , x 2 , . . . be formal variables, and set x 0 def = 1. We define the weight of π ∈ NC (m) For example, the partition displayed in Figure 1a has weight x 3 1 x 2 2 , the one in Figure 1b has weight x 1 x 3 2 . Moreover, we write |π| for the size of π, i.e., for the number of elements of the set which is partitioned by π.
We first study the case t = 1, and define where, again by definition, NC (m) 0 consists of just one element, namely the empty partition. Moreover, we define the generating function of m-divisible non-crossing partitions according to weight by (2.1) Proof. Node 1 must be in one of the blocks, say in a block of size mi. If v 1 = 1, v 2 , . . . , v mi are all the nodes in this block in linear order, then the nodes between v j and v j+1 , for j ∈ [mi], (where we identify v mi+1 with v 1 ) are involved in a smaller m-divisible non-crossing partition. Equation (2.1) then follows by standard generating function calculus: the block containing 1 contributes the term x i z mi , and the "small" m-divisible non-crossing partitions between v j and v j+1 each contribute a term C (m) (z) for j ∈ [mi].
Before we can move on to the case t > 1 we need to recall the Lagrange Inversion formula.
Lemma 2.6. Let f (z) be a formal power series with f (0) = 0 and f ′ (0) = 0, and let F (z) be its compositional inverse. Then, for all integers a and b,

2)
and is not easy to find explicitly in standard books. However, it can be derived without great effort by "partial integration" from (2.3): The coefficient of z −1 in the derivative of a Laurent series must necessarily be zero, which explains the third equality above.
From now on, let and let f (z) denote its compositional inverse. Slightly rewriting (2.1), we see that it is equivalent to Thus we have zC (m) (z) = f (z).
Lemma 2.7. For m, n, t ≥ 1 the generating function π∈NC (m) , the element j is contained in a block of size mi j of π, for some i j ≥ 1. These blocks contribute a weight of x i j z mi j , for j ∈ [t], to the generating function of m-divisible non-crossing t-partitions by weight.
Similar to the argument in the proof of Lemma 2.5, between the elements of these blocks in the range {t+1, . . . , mn} we find m(i 1 + · · · + i t ) − t + 1 "small" m-divisible non-crossing t-partitions, each contributing a term C (m) (z) to the generating function. We thus have If we use (2.2) with b = m(i 1 + · · · + i t ) − t + 2, a = −mn + t − 1 and the roles of f and F interchanged, then this expression becomes which is equivalent to the expression in the statement of the lemma.

Rank enumeration in NC
Furthermore, the number of m-divisible non-crossing partitions in NC (m) and the total number of m-divisible non-crossing partitions in NC (m) Proof. For proving (3.1), we must extract the coefficient of .
To the last term, we apply the easily verified fact Thus, the above expression is turned into As we said in the beginning, we must extract the coefficient of (3.4) in this expression. There is a single summand (in the sum over ℓ above) where we find this term: the one with ℓ = b 1 + b 2 + · · · + b n − t, and its coefficient is (3.1).
In order to establish (3.2) we use Corollary 2.2 to see that the rank of π ∈ NC (m) n;t with block structure described in the statement of the theorem is Thus, what we have to do is to set all x i 's equal to x and extract the coefficient of x n−s in (2.5). We start with the equivalent expression (3.5), with the x i 's specialised to x: The coefficient of x n−s in this expression is This is equivalent to the expression in (3.2). In order to obtain (3.3), one has to sum this expression over all s using the Chu-Vandermonde summation formula, see e.g., [19,Section 1,(5.27)], which is then followed by some simplification.
Next we compute the generating function for the completions of elements in NC (m) n;t by multi-chains of fixed length.
Proposition 3.2. Let l ≥ 2 be a positive integer, and let s ′ 2 , s 3 , . . . , s l+1 be non-negative where the sum is over multi-chains If l = 2, the empty product mn s 3 · · · mn s l of binomial coefficients has to be interpreted as 1.
Proof. We prove the assertion by induction on l.
For the start of the induction, we let l = 2 and we return to (2.5), which provides the generating function for m-divisible non-crossing partitions π 2 in NC (m) n;t . We consider the relation π 1 ≤ ref π 2 . How does π 1 arise from π 2 ? Clearly, what we have to do is to split some of the blocks of π 2 (possibly all) into smaller blocks to obtain π 1 . More precisely, each block of π 2 of size mb is replaced by an m-divisible non-crossing partition of mb elements. Hence, to model this in the generating function, we will replace in (2.5) the variable x b by the coefficient of u mb in xC (m) (u). (Recall that C (m) (u) is the generating function for m-divisible non-crossing partitions.) Here, the variable x keeps track of the number of blocks of π 2 and, thus, by the equation n − bl(π 2 ) = rk(π 2 ) = s ′ 2 , respectively equivalently, bl(π 2 ) = s 3 + t of the rank of π 2 . From the result we will then extract the coefficient of x s 3 +t to obtain the generating function (3.8).
Now, the substitution of u mb xC (m) (u) for x b , for b ≥ 1, in the series F (z) yields the expression Therefore, doing this substitution in (2.5), we arrive at Extracting the coefficient of x s 3 +t , we obtain At this point, we use again (3.6) to see that the above expression can be rewritten as which is seen to be equivalent to (3.9) with l = 2, once we use (2.3) with a = k and b = −mn − k and subsequently replace the summation index k by k + 1.
Next we perform the induction step. We assume that the assertion is true for multichains consisting of l−1 elements and consider the multi-chain The induction hypothesis implies that the generating function where the sum is over all multi-chains π 2 ≤ ref · · · ≤ ref π l of m-divisible non-crossing t-partitions, where the rank of π i is (s ′ 2 + s 3 ) + s 4 + · · · + s i , for i ∈ {3, . . . , l}, is given by We now consider the relation π 1 ≤ ref π 2 . We already saw that, in order to model this relation in the generating function, we have to replace the variable x b by the coefficient of u mb in xC (m) (u) everywhere. Again, the variable x keeps track of the number of blocks of π 2 . By Corollary 2.2 we have n − bl(π 2 ) = rk(π 2 ) = s ′ 2 , respectively equivalently, bl(π 2 ) = s 3 + s 4 + · · · + s l+1 + t.
Hence, from the result of the substitution we will then extract the coefficient of x s 3 +s 4 +···+s l+1 +t to obtain the generating function (3.8).
Using (3.10), we see that the substitution of The coefficient of x s 3 +s 4 +···+s l+1 +t in this expression equals (3.12) At this point we should note that the sums over k and ℓ have become completely independent. Moreover, the sum over k can be evaluated by means of the Chu-Vandermonde summation formula. Namely, we have .
If we substitute this in (3.12) and use (2.3) with a = ℓ and b = −mn − ℓ, and subsequently replace the summation index k by k + 1, then we obtain exactly (3.9).
We are now in the position to prove the rank enumeration with prescribed block structure of the first element in the multi-chain. Theorem 3.3. Let m, n, t, l be positive integers, and let s 1 , s 2 , . . . , s l+1 , b 1 , b 2 , . . . , b n be non-negative integers with s 1 + s 2 + · · · + s l+1 = n − t. The number of multi-chains π 1 ≤ ref π 2 ≤ ref · · · ≤ ref π l in the poset of m-divisible non-crossing t-partitions with the property that rk(π i ) = s 1 + s 2 + · · · + s i , for i ∈ [l], and that the number of blocks of size mi of π 1 is b i , for i ∈ [n], is given by if b 1 + 2b 2 + · · · + nb n = n and s 1 + b 1 + b 2 + · · · + b n = n, and 0 otherwise.
Proof. We use Proposition 3.2 with s ′ 2 = s 1 + s 2 . We have n − bl(π 1 ) = rk(π 1 ) = s 1 , or equivalently, bl(π 1 ) = s 2 + s 3 + · · · + s l+1 + t. Hence, we must replace x b by xx b in (3.9) and extract the coefficient of Here we have to extract the coefficient of x s 2 +s 3 +···+s l+1 +t . This is The sum over k is completely analogous to the sum over k in the proof of Proposition 3.2 and, hence, can as well be evaluated by means of the Chu-Vandermonde summation formula. In this way, the above expression turns into x i z mi s 2 +s 3 +···+s l+1 +t . (3.14) Now we must extract the coefficient of or zero if s 2 + s 3 + · · · + s l+1 + t = b 1 + b 2 + · · · + b n . However, our assumptions do indeed imply this relation. Little manipulation then leads to the expression (3.13).
We may now embark on the proof of Theorem 1.1.
Proof of Theorem 1.1. We reuse (3.14). Since, at this point, all rank conditions of the multi-chain π 1 ≤ ref π 2 ≤ ref · · · ≤ ref π l are already built in the calculation, it suffices to put all the x i 's equal to 1 and then finish the calculation. Consequently, we get Now the extraction of the coefficient of z ms 1 in the binomial series and little simplification finishes the proof.
We record two corollaries of Theorem 1.1. The first corollary provides a simple formula for the number of maximal chains in NC  Proof. Choose l = n − t, s 1 = s 2 = · · · = s n−t = 1, and s n−t+1 = 0 in Theorem 1.1.
The second corollary allows us to compute the zeta polynomial of NC   Proof. In Theorem 1.1, we replace l by l − 1. What we have to do is to sum the expression (1.1) over all possible s 1 , s 2 , . . . , s l . We have Both multiple sums are iterated Chu-Vandermonde convolutions. Thus, we obtain t n n + (l − 1)mn − t + 1 which can be simplified to the right-hand side of (3.15).    m) n;t µ(π 1 , π 2 )x rk(π 1 ) y rk(π 2 ) , (4.1)

The M-triangle of NC
where µ denotes the Möbius function of NC Proof. We follow a strategy that has been applied earlier in [22,Section 8] and [25,Section 9], and which exploits the equality µ(π 1 , π 2 ) = Z (m) n;t (π 1 , π 2 ; −1), where Z (m) n;t (π 1 , π 2 ; z) is the polynomial whose evaluation at positive z yields the number of multi-chains of length z − 1 which lie weakly between π 1 and π 2 . This equality is [36, Proposition 3.12.1(c)] tailored to our current situation.
In order to compute the coefficient of x r y s of M (m) n;t , we sum the zeta polynomials Z (m) n;t (π 1 , π 2 ; z) over all π 1 , π 2 ∈ NC (m) n;t with rk(π 1 ) = r and rk(π 2 ) = s, and evaluate the resulting expression at z = −1.
The zeta polynomials we require can be obtained from Theorem 1.1 by setting l = z + 1, s 1 = r, n − t − s l+1 = s, s 2 + s 3 + · · · + s l = s − r, and then summing over all possible s 2 , s 3 , . . . , s l . By using the Chu-Vandermonde summation formula, one obtains If we evaluate this expression at z = −1, then we obtain The main purpose of our consideration of the M-triangle of NC (m) n;t is a surprising connection conjectured in the case t = 1 by Chapoton in [12,13] for m = 1, and generalised by Armstrong in [1, Section 5.3] to m > 1.
This connection predicts the existence of two polynomials, denoted by F (m) n;t and H (m) n;t , with non-negative integer coefficients, that can be obtained from M (m) n;t by certain substitutions of the variables. In the case t = 1, explicit combinatorial explanations of these polynomials (as generating functions of certain combinatorial objects) are known. For t > 1, we conjecture a combinatorial description of one of these polynomials in Section 5.   n;t ∈ Z[x, y] with non-negative integer coefficients such that the following equalities hold: Proof. By Theorem 4.2, we have The sum over s can be evaluated by means of the Chu-Vandermonde summation formula. This yields Now the sum over r can be evaluated by means of the Chu-Vandermonde summation formula. Consequently, we obtain H (m) Using the restriction h ≤ n − t − k, we see that Since t ≥ 1 by assumption and k ≤ n−t due to the restriction on the first sum, the above expression is evidently positive. This shows that the coefficients of this polynomial are indeed positive integers. Now, using the expression for H The sum over h can be evaluated by means of the Chu-Vandermonde summation formula. If we additionally replace a by a − k, then we obtain Here, the sums over k can be evaluated by means of the Chu-Vandermonde summation formula. Thus, we arrive at The coefficient of x a y b in F   n;t the F -triangle and the H-triangle, respectively. Since Theorem 4.3 states that these polynomials have non-negative integer coefficients, it is an intriguing challenge to explain these polynomials combinatorially.
For t = 1, such explanations were given in [1,Section 5.3], generalising the case m = 1 from [12,13]. The F -triangle is the generating function for facets of the mdivisible cluster complex, where x marks the coloured positive roots, and y marks the coloured negative simple roots. The H-triangle is the generating function for positive chambers in the extended Shi arrangement, where x marks coloured floors and y marks coloured ceilings.
Starting from these concrete definitions of the F -, H-, and M-triangle, it is far from obvious that the relations from Theorem 4.3 hold; they were shown to be satisfied in several papers [3,22,23,25,37,38]. (These papers address a more general definition of those triangles for finite Coxeter groups.) Surprisingly, [18] defines F -, H-, and M-triangles in the context of yet another generalisation of non-crossing partitions. Conjecture 1.2 and Corollary 5.5 in [18] suggest that the relations from Theorem 4.3 should hold in their setting, too.
be the set of ordered pairs of integers between 1 and n. We define a partial order on T n by setting (i, j) (k, l) if and only if i ≥ k and j ≤ l (5 .1) for all (i, j), (k, l) ∈ T n . The poset (T n , ) is the triangular poset of degree n. For (i, j), (k, l) ∈ T n we define their formal sum by which extends to subsets of T n as follows: A filter (of T n ) is a set X ⊆ T n such that (i, j) ∈ X and (i, j) (k, l) together imply (k, l) ∈ X. A multi-chain of filters (of T n ) is a tuple (V m , V m−1 , . . . , V 1 ) of filters with V m ⊆ V m−1 ⊆ · · · ⊆ V 1 ⊆ T n . The next definition is adapted from [2, page 180].
holds for all indices i, j ≥ 1 (where we set V k = V m for k > m), and holds for all i, j ≥ 1 with i + j ≤ m.
Now, for t ∈ [n], consider the filter A t-filter is a filter of the poset (T n;t , ), and a multi-chain of t-filters is a tuple (V m , V m−1 , . . . , V 1 ) of t-filters with V m ⊆ V m−1 ⊆ · · · ⊆ V 1 . By definition, every multi-chain of t-filters is a multi-chain of filters. We call a multichain of t-filters geometric if it satisfies both (5.2) and (5.3).

Conjecture 5.2. The number of geometric multi-chains of t-filters
In fact, it may appear more natural to generalise the notion of geometric multichains to T n;t by adapting (5.3) so that we take complements with respect to T n;t . This definition, however, does not produce the desired number of geometric multi-chains as the next example shows. The number that appears in Conjecture 5.2 is desirable, because it is precisely the cardinality of NC         Proof. For t = 1, the claim follows from [2, Theorem 3.6], which states that the geometric multi-chains of filters correspond bijectively to regions in the fundamental chamber of the extended Shi arrangement. Moreover, the adjacency graph of these regions corresponds to the poset diagram of NN (m) n , ⊆ . Then, crossing a wall in the fundamental chamber corresponds to a covering pair in NN (m) n;t , ⊆ ; the colour of this wall determines the index j from the statement, and the normal vector of the hyperplane supporting this wall determines the unique element in which the j-th components of the multi-chains in question differ.
For t > 1, the claim follows from the fact that T n;t is a filter of T n , which implies that covering pairs in NN Example 5.7. Let us continue Example 5.3. Figure 3 shows NN  n;t consists simply of the t-filters of T n;t . We will think of these filters equivalently as certain lattice paths.
A Dyck path is a lattice path starting at the origin, ending on the x-axis, and which consists only of steps of the form (1, 1) (up-steps) and (1, −1) (down-steps) while never going below the x-axis. The length of a Dyck path P is its number of steps, denoted by ℓ(P ). A t-Dyck path is a Dyck path that starts with t up-steps. Let D n;t denote the set of all t-Dyck paths of length 2n.
A valley of a Dyck path is a coordinate on this path which is preceded by a down-step, and followed by an up-step.       A peak of a Dyck path is a coordinate on this path which is preceded by an up-step, and followed by a down-step. We note that the set of peaks (or respectively, valleys) uniquely determines a Dyck path. The height of a peak (or respectively a valley) is its ordinate. For P ∈ D n;t , we write v(P ) for the number of valleys of P , we write p(P ) for the number of peaks of P , and we write r(P ) for the number of valleys of P of height 0.
For P 1 , P 2 ∈ D n;t we say that P 1 dominates P 2 if P 2 lies weakly below P 1 (as a lattice path). We denote 3 this relation by P 1 ≤ ddom P 2 . The partially ordered set D n;t , ≤ ddom is isomorphic to NN (1) n;t , ⊆ via the bijection from Lemma 5.8, and its dual was for instance studied in [5,7,17,27,28]. Figure 5 shows D 4;2 , ≤ ddom . x v(P ) y r(P ) . (5.7) Proof. Observe that, since we are in the case m = 1, the empty set is not in the range of the map fl 1 . Now let W ∈ NN (1) n;t and Q = Θ(W) ∈ D n;t be the t-Dyck path corresponding to W via the bijection from Lemma 5.8.
By construction, if V is covered by W, then there exists a unique valley of Q, say at coordinate (p, q), such that Θ(V) agrees with Q except that it runs through (p, q+2) instead of (p, q). More precisely, the path Θ(V) arises from Q by changing the down-step before (p, q) into an up-step, and by changing the up-step after (p, q) into a down-step. This establishes FL 1 (W) = v(Q). Now, let V be an element in NN (1) n;t , ⊆ covered by W, and let P = Θ(V) be the corresponding t-Dyck path. Again by construction, we find that fl 1 (V, W) ∈ S n;t if and only if P has a peak (p, 2) and Q has a valley (p, 0). Therefore FL 1 (W) ∩ S n;t = r(Q). In Figure 5, we have labelled the elements of the poset additionally by the term they contribute toH (1) 4;2 . The readers may convince themselves that Proposition 5.10 holds. We obtainH (1) 4;2 (x, y) = x 2 y 2 + x 2 y + x 2 + 2xy + 3x + 1, which confirms Conjecture 5.6 in this case.
Remark 5.11. The combinatorial description ofH (1) n;t (x, y) described in Proposition 5.10 was found with the help of FindStat [33]. In [29,Section 5], a combinatorial description of H (1) n;t (x, y) is conjectured, which counts t-Dyck paths according to various kinds of peaks. We leave it as a challenge for the readers to give a bijection on D n;t that exchanges the corresponding pairs of statistics.
Before we conclude this article with the proof of Conjecture 5.6 in the case m = 1, we recall the Lagrange-Bürmann formula.
Proof. We first determine the generating function for (all) Dyck paths, where z keeps track of the length and x keeps track of the number of valleys, see (5.8) below. Based on that equation, we then determine the generating function for t-Dyck paths, where again z keeps track of the length and x keeps track of the number of valleys, but where also y keeps track of the valleys at height 0, see (5.11).
A Dyck path can either be empty, it may have no valleys of height 0, or it may have a valley of height 0. In the latter case, such a Dyck path may be decomposed as uP 1 dP 2 , where u denotes an up-step, d denotes a down-step, and P 1 and P 2 stand for Dyck paths, of which P 1 can possibly be empty.
Let us abbreviate x v(P ) z ℓ(P )/2 denote the generating function for Dyck paths by the number of valleys. According to the reasoning in the first paragraph of this proof, we obtain the equation Equivalently, writing D(x; z) def = D 1 (x; z) − 1, this can be rewritten as In fact, we want to consider t-Dyck paths, which are precisely the Dyck paths that start with t up-steps. Let us write and D 1,t (x, y; z) def = P ∈D•;t x v(P ) y r(P ) z ℓ(P )/2 .
A t-Dyck path may be decomposed in the form for some non-negative integer s, where the P i 's andP i 's again stand for (possibly empty) Dyck paths. In D 1,t (x, y; z), the path P 1 contributes a factor zD 1 (x; z), because it cannot be preceded by a valley. If i > 1, then the path P i is preceded by a valley (of height > 0) if and only if it is not empty, and therefore contributes a factor z 1 + x D 1 (x; z) − 1 . The collection of theP i 's forms an ordinary Dyck path separated by its valleys of height 0, and therefore, this piece contributes a factor ∞ s=0 .
We have just explained the following form for the generating function: Proposition 5.10 then implies By (5.9), the compositional inverse of D(x; z) is The "running" denominator n − t − s has cancelled! We see by inspection that for r = s = n − t the summand in the above double sum equals (xy) n−t , which is exactly the separate term in the above expression. We may therefore integrate it into the double sum. Consequently, we havẽ In the final step, we replace r by n − t − r and s by n − t − r − s. Thereby, we arrive at where the last line follows from the expression for H (m) n;t (x, y) that we derived in (4.3).
Remark 5.14. By construction, it is obvious that multi-chains of t-filters correspond to multi-chains of t-Dyck paths with respect to ≤ ddom . It remains to understand how Conditions (5.2) and (5.3) translate to t-Dyck paths.

An extension to Coxeter groups
In this last section, we outline a possible construction of F -, H-and M-triangles, for m = 1, in the setting of parabolic quotients of finite Coxeter groups. We wish to keep this part brief and explain this extension only in the case of the symmetric group. The general definitions for finite Coxeter groups can be found in [31,Section 6] or [39,Chapter 5].
We fix a composition α = (α 1 , α 2 , . . . , α r ) of n > 0 and colour a collection of nodes labelled by 1, 2, . . . , n according to the components of α, that is, the nodes 1, 2, . . . , α 1 are assigned the first colour, the nodes α 1 +1, . . . , α 1 +α 2 are assigned the second colour, etc. An α-partition is a set partition of [n] for which no block contains two nodes of the same colour. The arc diagram of an α-partition π is obtained as follows: if i and j are adjacent members of some block of π, then we connect the nodes labelled by i and j by an arc which leaves the node i (which has colour c i , say) to the bottom, passes below all nodes of colour c i , passes above subsequent nodes, and finally enters the node labelled j from above. An α-partition is non-crossing if its diagram can be drawn such that no two arcs cross. Let NC α denote the set of all non-crossing α-partitions. Figure 6 shows the refinement order on NC (2,2) . Clearly, if α = (t, 1, 1, . . . , 1) is a composition of n, then NC α = NC (1) n;t . We may thus define the M-triangle of NC α , denoted by M α (x, y), analogously to Definition 4.1. By inspection of Figure 6, we obtain M (2,2) (x, y) = x 2 y 2 − 2xy 2 + 4xy + y 2 − 4y + 1.
Note that H (2,2) (x, y) has a negative coefficient, and can therefore not arise in the spirit of Proposition 5.10 as the sum over some lattice paths with respect to certain statistics.
We observe that the polynomialM (2,1,2) (x, y) cannot arise from some graded poset P in the manner described in Definition 4.1. Indeed, if this were the case, then the constant term ofM (2,1,2) (x, y) forces P to have a unique minimal element which must be covered by nine elements, because the coefficient of y is −9. However, the coefficient of xy is 8, implying that P has only eight elements of rank 1 which is a contradiction.
The previous examples show that the correspondence between F -, H-and M-triangles is not so well-behaved for arbitrary parabolic quotients of the symmetric group. Conjectures 5.3 and 5.4 in [30] claim that this correspondence holds precisely when α has at most one component larger than 1. Computer experiments suggest that this behaviour carries over similarly to Coxeter groups of other types. Remarkably, the various Ftriangles so obtained seem to always have non-negative coefficients, maybe hinting at interesting combinatorics to be discovered.