The Rise-Contact involution on Tamari intervals

We describe an involution on Tamari intervals and m-Tamari intervals. This involution switches two sets of statistics known as the"rises"and the"contacts"and so proves an open conjecture from Pr\'eville-Ratelle on intervals of the m-Tamari lattice.


Introduction
The Tamari lattice [Tam62,HT72] is a well known lattice on Catalan objects, most frequently described on binary trees, Dyck paths, and triangulations of a polygon. Among its many interesting combinatorial properties, we find the study of its intervals. Indeed, it was shown by Chapoton [Cha07] that the number of intervals of the Tamari lattice on objects of size n is given by (1.1) 2 n(n + 1) This is a surprising result. Indeed, it is not common that we find a closed formula counting intervals in a lattice. For example, there is no such formula to count the intervals of the weak order on permutations.
Even more surprising is that this formula also counts the number of simple rooted triangular maps which led Bernardi and Bonichon to describe a bijection between Tamari intervals and said maps [BB09]. This is a strong indication that Tamari intervals have deep and interesting combinatorial properties.
One generalization of the Tamari lattice is to describe it on m-Catalan objects. This was done by Bergeron and Préville-Ratelle [BPR12]. Again, they conjectured that the number of intervals could be counted by a closed formula which was later proved in [BMFPR11]: (1.2) m + 1 n(mn + 1) (m + 1) 2 n + m n − 1 .
In this case, the connection to maps is still an open question. In their paper, the authors of [BMFPR11] noticed an equi-distribution on Tamari intervals between two statistics related to contacts and rises of the involved Dyck paths. At this stage, the equi-distribution could be seen directly on the generating function of the intervals but there was no combinatorial explanation. In his thesis [PR12], Préville-Ratelle developed the subject and left some open problems and conjectures. The one related to the contacts and rises of Tamari intervals is Conjecture 17 and this the one we propose to prove in this paper. It describes a equi-distribution not only between two statistics (as in [BMFPR11]) but between two sets of statistics. Basically, in [BMFPR11], only the initial rise of a Dyck path was considered whereas in Conjecture 17, Préville-Ratelle considers all positive rises of the Dyck path. Besides, a third statistic is described, the distance which also appears in many other open conjectures and problems of Préville-Ratelle 's thesis: it is related to trivariate diagonal harmonics which is the original motivation of the m-Tamari lattice. According to Préville-Ratelle, Conjecture 17 can be proved both combinatorially 1 and through the generating function when m = 1. But until now, there was no proof of this result when m > 1.
To prove this conjecture, we use a combinatorial object that we introduced in a previous paper on Tamari intervals [CP15]: the intervalposets. They are posets on integers, satisfying some simple local rules, and are in bijections with the Tamari intervals. Besides, their structure includes two planar forests (from the two bounds of the Tamari interval) which are very similar to the Schnyder woods of the triangular planar maps. Another quality of interval-posets is that m-Tamari intervals are also in bijection with a sub family of interval-posets which was the key to prove the result when m > 1.
Section 2 of this paper gives a proper definition of Tamari intervalposet and re-explore the link with the Tamari lattice in the context of our problem. In Section 3, we describe the rise, contact, and distance statistics and their relations to interval-posets statistics. This allows us to state Theorem 3.4 which expresses our version of Conjecture 17 in the case m = 1. Section 4 is dedicated to the proof of Theorem 3.4 through an involution on interval-posets described in Theorem 4.22. However, the main results of our paper lies in our last section, Section 5, where we are able to generalize the involution to the m > 1 case. Theorem 5.5 is a direct reformulation of Conjecture 17 from [PR12]. It is a consequence of Theorem 5.18 which describes an involution on intervals of the m-Tamari lattice.
Remark 1.1. A previous version of this involution was described in a extended abstract [CCP14]. This was only for the m = 1 case and did not include the whole set of statistics. Also, in this original description, the fact that it was an involution could be proved but was not clear. We leave it to the curious reader to see that the bijection described in [CCP14] is indeed the same as the one we are presenting in details now.
Remark 1.2. This paper comes with a complement Sage-Jupyter notebook [Pon] available on github and binder. This notebook contains Sage code for all computations and algorithms described in the paper. The binder system allows the reader to run and edit the notebook on line.
2. Tamari Interval-posets 2.1. Definition. Let us first introduce some notations that we will need further on. In the following, if P is a poset, then we denote by ⊳ P , P , ⊲ P and P the smaller, smaller-or-equal, greater and greateror-equal, respectively, relations of the poset P . When the poset P can be uniquely inferred from the context, we will sometimes leave out the subscript "P ". We write (2.1) rel(P ) = {(x, y) ∈ P, x ⊳ y} for the set of relations of P . A relation (x, y) is said to be a cover relation if there is no z in P such that x ⊳ z ⊳ y. The Hasse diagram of a poset P is the directed graph formed by the cover relations of the poset. A poset is traditionally represented by its Hasse diagram.
We say that we add a relation (i, j) to a poset P when we add (i, j) to rel(P ) along with all relations obtained by transitivity (this requires that neither i ⊳ P j nor j ⊳ P i before the addition). Basically, this means we add an edge to the Hasse Diagram. The new poset P is then an extension of the original poset.
We now give a first possible definition of interval-posets.
Definition 2.1. A Tamari interval-poset (simply referred as intervalposet in this paper) is a poset P on {1, 2, ..., n} for some n ∈ N, such that all triplets a < b < c in P satisfies the following property which we call the Tamari axiom: Figure 1 shows an example and a counter-example of interval-posets. The first poset is indeed an interval-poset. The Tamari axiom has to be checked on every a < b < c such that there is a relation between a and c: we check the axiom on 1 < 2 < 3 and 3 < 4 < 5 and it is satisfied. The second poset of Figure 1 is not an interval poset: it contains 1 ⊳ 3 but not 2 ⊳ 3 so the Tamari axiom is not satisfied for 1 < 2 < 3. Definition 2.2. Let P be an interval-poset and a, b ∈ P such that a < b. Then • if a ⊳ b, then (a, b) is said to be an increasing relation of P .
• if b ⊳ a, then (b, a) is said to be a decreasing relation of P .
As an example, the increasing relations of the interval-poset of Figure 1 are (1, 3) and (2, 3) and the decreasing relations are (2, 1), (4, 3), and (5, 3). Clearly a relation x ⊳ y is always either increasing or decreasing and so one can split the relations of P into two non-intersecting sets.
Definition 2.3. Let P be an interval-poset. Then, the final forest of P , denoted by F ≥ (P ), is the poset formed by the decreasing relations of P , i.e., b ⊳ F ≥ (P ) a if and only if (b, a) is a decreasing relation of P . Similarly, the initial forest of P , denoted by F ≤ (P ), is the poset formed by the increasing relations of P .
By Definition 2.1 it is immediate that the final and initial forests of an interval-poset are also interval-posets. By extension, we say that an interval-poset containing only decreasing (resp. increasing) relations is a final forest (resp. initial forest). The designation forest comes from the result proved in [CP15] that an interval-poset containing only increasing (resp. decreasing) relations has indeed the structure of a planar forest, i.e., every vertex in the Hasse diagram has at most one outgoing edge.
The increasing and decreasing relations of an interval-poset play a significant role in the structure and properties of the object. We thus follow the convention described in [CP15] to draw interval-posets which differs from the usual representation of posets through their Hasse diagram. Indeed, each interval-poset is represented with an overlay of the Hasse Diagrams of both its initial and final forests. By convention, an increasing relation b ⊳ c with b < c is represented in blue with c on the right of b. A decreasing relation b ⊳ a with a < b is represented in red with a above b. In general a relation (either increasing or decreasing) between two vertices x ⊳ y is always represented such that y is on a righter and upper position compared to x. Thus, the color code, even though practical, is not essential to read the figures. Figure 2 shows the final and initial forests of the interval-poset of Figure 1. A more comprehensive example is shown in Figure 3. Following our conventions, you can read off, for example, that 3 ⊳ 4 ⊳ 5 and that 9 ⊳ 8 ⊳ 5.
Hasse diagram of P F ≥ (P ) F ≤ (P ) P drawn as interval-poset We also define some vocabulary on the vertices of the interval-posets related to the initial and final forests.
Definition 2.4. Let P be an interval-poset. Then • a vertex b is said to be a decreasing root of P if there is no a < b with a decreasing relation b ⊳ a; • a vertex b is said to be an increasing root of P if there is no c > b with an increasing relation b ⊳ c; • a decreasing-cover (resp. increasing-cover) relation is a cover relation of the final (resp. initial) forest of P ; • the decreasing children of a vertex b are all elements c > b such that c ⊳ b is a decreasing-cover relation; • the increasing children of a vertex b are all elements a < b such that a ⊳ b is a increasing-cover relation.
We also need to refine the notion of extension related to increasing and decreasing relations.
In other words, J is an extension of I if it is obtained by adding relations to I, it is a decreasing-extension if it is obtained by adding only decreasing relations and it is a increasing-extension if it is obtained by adding only increasing relations.
Remark 2.6. If you add a decreasing relation (b, a) to an intervalposet I, all extra relations that are obtained by transitivity are also decreasing. Indeed, suppose that J is obtained from I by adding the relation b ⊳ a with a < b (in particular neither (a, b) nor (b, a) is a relation of I). And suppose that the relation i ⊳ J j with i < j is added by transitivity which means i ⋪ I j, i I b and a I j. If i < a, the Tamari axiom on (i, a, b) implies a ⊳ I b which contradicts our initial statement. So we have a < i < j and a ⊳ I j, the Tamari axiom on (a, i, j) implies i ⊳ I j and again contradicts our statement. Note on the other hand that nothing guarantees that the obtained poset is still an interval-poset. Similarly, if you add an increasing relation (a, b) to an interval-poset, you obtain an increasing-extension.
2.2. The Tamari lattice. It was shown in [CP15] that Tamari intervalposets are in bijection with intervals of the Tamari lattice. The main purpose of this paper is to prove a conjecture of Préville-Ratelle [PR12] on Tamari intervals. To do so, we first give a detailed description of the relations between interval-posets and the realizations of the Tamari lattice in terms of trees and Dyck paths. Let us start with some reminder on the Tamari lattice.

Definition 2.7. A binary tree is recursively defined by being either
• the empty tree, denoted by ∅, • a pair of binary trees, respectively called left and right subtrees, grafted on a node. If L and R are two binary trees, we denote by •(L, R) the binary tree obtained from L and R grafted on a node.
What we call a binary tree is often called a planar binary tree in the literature (as the order on the subtrees is important). Note that in our representation of binary trees, we never draw the empty subtrees.
The size of a binary tree is defined recursively: the size of the empty tree is 0, and the size of a tree •(L, R) is the sum of the sizes of L and R plus 1. It is also the number of nodes. For example, the following tree has size 3, it is given by the recursive grafting •(•(∅, •(∅, ∅)), ∅). It is well known that the unlabeled binary trees of size n are counted by the n th Catalan number (2.2) 1 n + 1 2n n .
Definition 2.8 (Standard binary search tree labeling). Let T be a binary tree of size n. The binary search tree labeling of T is the unique labeling of T with labels 1, . . . , n such that for a node labeled k, all nodes on the left subtree of k have labels smaller than k and all nodes on the right subtree of k have labels greater than k. An example is given in Figure 4.
In other words, the binary search tree labeling of T is an in-order recursive traversal of T : left, root, right. For the rest of the paper, we identify binary trees with their corresponding binary search tree 1 2 3 4 5 Figure 4. A binary search tree labeling labeling. In particular, we write v 1 , . . . , v n the nodes of T : the index of the node corresponds to its label in the binary search tree labeling.
To define the Tamari lattice, we need the following operation on binary trees. Definition 2.9. Let v y be a node of T with a non-empty left subtree of root v x . The right rotation of T on v y is a local rewriting which follows Figure 5 Figure 5. Right rotation on a binary tree.
It is easy to check that the right rotation preserves the binary search tree labeling. It is the cover relation of the Tamari lattice [Tam62, HT72]: a binary tree T is said to be bigger in the Tamari lattice than a binary tree T ′ if it can be obtained from T ′ through a sequence of right rotations. The lattices for the sizes 3 and 4 are given in Figure 6.
Dyck paths are another common set of objects used to define the Tamari lattice. First, we recall their definition.
Definition 2.10. A Dyck path of size n is a lattice path from the origin (0, 0) to the point (2n, 0) made from a sequence of up-steps (steps of the form (x, y) → (x + 1, y + 1)) and down-steps (steps of the form (x, y) → (x + 1, y − 1)) such that the path stays above the line y = 0.
A Dyck path can also be considered as a binary word by replacing up-steps by the letter 1 and down-steps by 0. We call a Dyck path primitive if it only touches the line y = 0 on its end points. As widely known, Dyck paths are also counted by the Catalan numbers. There are many ways to define a bijection between Dyck paths and binary trees. The one we use here is the only one which is consistent with the usual definition of the Tamari order on Dyck paths. Definition 2.11. We define the tree map from the set of all Dyck paths to the set of binary trees recursively. Let D be a Dyck path.
• If D is empty, then tree(D) to be the empty binary tree.
• If D is of size n > 0, then the binary word of D can be written uniquely as D 1 1D 2 0 where D 1 and D 2 are Dyck paths of size smaller than n (in particular, they can be empty paths). Then tree(D) is the tree •(tree(D 1 ), tree(D 2 )).
Note that the path defined by 1D 2 0 is primitive; it is the only nonempty right factor of the binary word of D which is a primitive Dyck path. Similarly, the subpath D 1 corresponds to the left factor of D up to the last touching point before the end. Consequently, if D is primitive, then D = 1D 2 0, while D 1 is empty and thus tree(D) is a binary tree whose left subtree is empty. If both D 1 and D 2 are empty, then D = 10, the only Dyck path of size 1, and tree(D) is the binary tree formed by a single node.
The tree map is a bijection and preserves the size as it is illustrated in Figure 7. Figure 7. Bijection between Dyck paths and binary trees.

←→
Following this bijection, one can check that the right rotation on binary trees corresponds to the following operation on Dyck paths. By extension, we then say that a Dyck path D is bigger than a Dyck path D ′ in the Tamari lattice if it can be obtained from D ′ through a series of right rotations. The Tamari lattices of sizes 3 and 4 in terms of Dyck paths are given in Figure 9.  An example of the construction is given in Figure 10. As explained in [CP15], both the initial and the final forest constructions give bijections between binary trees and planar forests, i.e., forests of trees where the order on the trees is fixed as well as the orders of the subtrees of each node. Indeed, first notice that the labeling on both images F ≥ (T ) and F ≤ (T ) is entirely canonical (such as the labeling on the binary tree) and can be retrieved by only fixing the order in which to read the trees and subtrees. Then these are actually well known bijections. The one giving the final forest is often referred to as "left child = left brother" because it can be achieved directly on the unlabeled binary tree by transforming every left child node into a left brother and by leaving the right child nodes as sons. Thus in Figure 10, 2 is the left child of 3 in T and it becomes the left brother of 3 in F ≥ (T ), 9 is a right child of 7 in T and it stays the right-most child of 7 in F ≥ (T ). The increasing forest construction is then the "right child = right brother" bijection.
Also, the initial and final forests of a binary tree T are indeed initial and final forests in the sense of interval-posets. In particular, they are interval-posets. The fact that they contain only increasing (resp. decreasing) relations is given by construction. It is left to check that they satisfy the Tamari axiom on all their elements: this is due to the binary search tree structure. In particular, if you interpret a binary search tree as poset by pointing all edges toward the root then it is an interval-poset.
Theorem 2.14 (from [CP15]). Let T 1 and T 2 be two binary trees and R = rel(F ≥ (T 1 )) ∪ rel(F ≤ (T 2 )). Then, R is the set of relations of a poset P if and only if T 1 ≤ T 2 in the Tamari lattice. And in this case, P is an interval-poset.
This construction defines a bijection between interval-posets and intervals of the Tamari lattice.
There are two ways in which R could be not defining a poset. First, R could be non transitive. Because of the structure of initial and final forests, this never happens. Secondly, R could be non anti-symmetric by containing both (a, b) and (b, a) for some a, b ≤ n. This happens if and only if T 1 ≤ T 2 . You can read more about this bijection in [CP15]. Figure 11 gives an example. To better understand the relations between Tamari intervals and interval-posets, we now recall some results from [CP15, Prop. 2.9] which are immediate from the construction of interval-posets and the properties of initial and final forests. As the Tamari lattice is also often defined on Dyck paths, it is legitimate to wonder what is the direct bijection between a Tamari interval [D 1 , D 2 ] of Dyck paths and an interval-poset. Of course, one can just transform D 1 and D 2 into binary trees through the bijection of Definition 2.11 and then construct the corresponding final and initial forests. But because many statistics we study in this paper are more naturally defined on Dyck paths than on binary trees, we give the direct construction.
Recall that for each up-step d in a Dyck path, there is a corresponding down-step d ′ which is the first step you meet by drawing a horizontal line starting from d. From this, one can define a notion of nesting: an up-step d 2 (and its corresponding down-step d ′ 2 ) is nested in (d, d ′ ) if it appears in between d, d ′ in the binary word of the Dyck path. This bijection is actually a very classical one. It consists of shrinking the Dyck path into a tree skeleton. In Figure 12, we show in parallel the process of Proposition 2.16 on the Dyck path and the corresponding binary tree. Step 1: label the up-steps and their corresponding down-steps from left to right. Step 2: transform nestings into poset relations.  Proof. We use the recursive definition of the tree map. Let D be a non empty Dyck path and T = tree(D). We want to check that P is equal to F := F ≥ (T ). The path D decomposes into D = D 1 1D 2 0 with tree(D 1 ) = T 1 the left subtree of T and tree(D 2 ) = T 2 , the right subtree of T . We assume by induction that the proposition is true on F ≥ (D 1 ) and F ≥ (D 2 ). Let 1 ≤ k ≤ n be such that size(D 1 ) = k − 1 (in Figure 12, k = 5): then k is the label of the pair (1, 0) which appears in the decomposition of D. We also have that v k is the root of T . Now let us choose a < b ≤ n. Either • a < b < k: the pairs of steps labeled by a and b both belong to D 1 , we have b ⊳ P a if and only if b ⊳ F a by induction. • b = k: the pair labeled by a belongs to D 1 . It does not nest k, so b ⋪ P a. In T , v a is in T 1 , the left subtree of T and so we also have b ⋪ F a. • a = k: the pair labeled by b belongs to D 2 . It is nested in k, so b ⊳ P a. In T , v b belongs to T 2 the right subtree of D, we have b ⊳ F a. • k < a < b: the pairs of steps labeled by a and b both belong to D 2 , we have b ⊳ P a if and only if b ⊳ F a by induction.
On binary trees, the constructions of the final and initial forests are completely symmetrical: the difference between the two only consists of a choice between left subtrees and right subtrees. Because the leftright symmetry of binary trees is not obvious when working on Dyck paths, the construction of the initial forest from a Dyck path gives a different algorithm than the final forest one.
Proposition 2.17. Let D be a Dyck path of size n, we construct a directed graph following this process: • label all up-steps of D from 1 to n from left to right, • for each up-step a, find, if any, the first up-step b following the corresponding down-step of a and add the edge a −→ b.
Then this resulting directed graph is the Hasse diagram of the initial forest of D.
The construction is illustrated on Figure 13.
Proof. We use the same induction technique as for the previous proof.
As before, we have D = D 1 1D 2 0 along with the corresponding trees T , T 1 , and T 2 and size(D 1 ) = size(T 1 ) = k − 1. We set F := F ≤ (T ) and we call P the poset obtained by the algorithm. Step 1: label all up-steps from left to right. Step 2: Connect each up-step to the first-up step following its down-step.  First, let us prove that for all a < k, we have a ⊳ P k. Indeed suppose there exists a < k with a ⋪ P k, we take a to be maximal among those satisfying these conditions. We have a ∈ D 1 so its corresponding down-step appears before k, let a ′ ≤ k be the first up-step following the down-step of a. If a ′ = k, then (a, k) is in the Hasse diagram of P and so a ⊳ P k. If a ′ < k, we have a ⊳ P a ′ by definition and the maximality of a gives a ′ ⊳ k which implies a ⊳ P k by transitivity. Now let us choose a < b ≤ n. Either • a < b < k: the up-steps labeled by a and b both belong to D 1 , we have a ⊳ P b if and only if a ⊳ F b by induction. • b = k: in T , b is the root and a is in its left subtree: we have a ⊳ F b. In P , we have also proved a ⊳ P b. • a = k: the corresponding down-step of a is the last step of D which means there is no edge (a, b) in P . Similarly, because a is the tree root, there is no edge (a, b) in F . • k < a < b: the up-steps labeled by a and b both belong to D 2 , we have a ⊳ P b if and only if a ⊳ F b by induction. Now that we have described the relation between interval-posets and Tamari intervals both in terms of binary trees and Dyck path, we will often identify a Tamari interval with its interval-poset. When we refer to Tamari intervals in the future, we consider that they can be given indifferently by a interval-poset or by a couple of a lower bound and an  . This is also true of C(D) even though it is less obvious. It will become clear once we express the statistics in terms of planar forests. At first, let us use the definitions on Dyck paths to express our main result on Tamari intervals.
Definition 3.2. Consider an interval I of the Tamari lattice described by two Dyck paths D 1 and To summarize, all the statistics we defined on Dyck paths are extended to Tamari intervals by looking at the lower bound Dyck path D 1 when considering contacts and the upper bound Dyck path D 2 when considering rises.
Most of these statistics have been considered before on both Dyck paths and Tamari intervals. In [BMFPR11], one can find the same definitions for the initial rise r 0 (I) and number of non-final contacts c 0 (I). Taking x 0 = y 0 = 1 in C(I, X) and R(I, Y ) corresponds to ignoring 0 values in C(I) and R(I): we find those monomials in Préville-Ratelle's thesis [PR12]. Our definition of C(I, X) is slightly different than the one of Préville-Ratelle: we will explain the correspondence in the more general case of m-Tamari intervals in Section 5. We now describe another statistic from [PR12] which is specific to Tamari intervals: it cannot be defined through a Dyck path statistics on the interval end points.
Definition 3.3. Let I = [D 1 , D 2 ] be an interval of the Tamari lattice. A chain between D 1 and D 2 is a list of Dyck paths which connects D 1 and D 2 in the Tamari lattice. If the chains comprises k elements, we say its of length k − 1 (the number of cover relations).
We call the distance of I and write d(I) the maximal length of all chains between D 1 and D 2 .
For example, if I = [D, D] is reduced to a single element, then d(I) = 0. If I = [D 1 , D 2 ] and D 1 ≤ D 2 is a cover relation of the Tamari lattice, then d(I) = 1. This statistic was first described in [BPR12], it generalizes the notion of area of a Dyck path to an interval. To finish, we need the notation size(I) which is defined to be the size of the elements of I: if I is an interval of Dyck paths of size n, then size(I) = n. Note that is is also the number of vertices of the intervalposet representing I.We can now state the first version of the main result of this paper.
For x 0 = y 0 = 1, this corresponds to a special case of [PR12, Conjecture 17] where m = 1, the general case will be dealt in Section 5. The case where X, Y, and q are set to 1 is proved algebraically in [BMFPR11]. In this paper, we give a combinatorial proof by describing an involution on Tamari intervals that switches c 0 and r 0 as well as C and R. The involution is described in Section 4. First, we need some interpretations of the statistics at hands in terms of interval-posets.
Definition 3.5. Let I be an interval-poset of size n, we define • dc 0 (I) (resp. ic ∞ (I)) is the number of decreasing (resp. increasing) roots of I. • dc i (I) (resp. ic i (I)) for 1 ≤ i ≤ n is the number of decreasing (resp. increasing) children of the vertex i. Note that we do not include dc n nor ic 1 in the corresponding vectors as they are always 0. The vertices of I are read in their natural order in DC and in reverse order in IC: this follows a natural traversal of the final (resp. initial) forests from roots to leaves. As an example, in Figure 3, we have DC(I) = (3, 0, 2, 0, 0, 4, 0, 0, 1, 0) and IC(I) = (4, 2, 0, 0, 1, 0, 2, 1, 0, 0). Proof. This is clear from the construction of the final forest from the Dyck path given in Proposition 2.16. Indeed, each non-final contact of the Dyck path corresponds to exactly one decreasing root of the interval-poset. Then the decreasing children of a vertex are the contacts of the Dyck path nested in the corresponding (up-step, down-step) tuple.
Remark 3.7. The vector IC(I) is not equal to R(I) in general. In fact, the interpretation of rises directly on the interval-poset is not easy. What we will prove anyway is that the two vectors can be exchanged through an involution on I. This involution is shown in Section 4 and is a crucial step in proving Theorem 3.4.

Distance and Tamari inversions.
Before describing the involutions used to prove Theorem 3.4, we discuss more the distance statistics on Tamari intervals in order to give a direct interpretation of it on interval-posets.
We write TInv(I) the set of Tamari inversions of a set I.
As an example, the Tamari inversions of the interval-poset of Figure 3 are exactly (1, 2), (1, 5), (7, 8), (9, 10). As counter examples, you can see that (1, 6) is not a Tamari inversion because we have 1 < 5 < 6 and 6 ⊳ 5. Similarly, (6, 8) is not a Tamari inversion because there is 6 < 7 < 8 and 6 ⊳ 7. Note also that if (a, b) is a Tamari inversion of I, then a ⋪ b and b ⋪ a. Our goal is to prove the following statement. The proof of Proposition 3.9 requires two inner results that we express as Lemmas. Proof. By Proposition 2.15, we know that I ′ is a decreasing-extension of I. This Lemma is then just a refinement of Proposition 2.15 which states that the decreasing relations that have been added come from the Tamari inversions of I.
Let (b, a) be a decreasing-cover relation of I ′ such that b ⋪ I a. Because I ′ is an extension of I, we also know that a ⋪ I b. Let k be such that a < k < b. Because we have b ⊳ I ′ a, the Tamari axiom on a, k, b gives us k ⊳ I ′ a. This implies that b ⋪ I ′ k as (b, a) is a decreasing-cover relation of I ′ by hypothesis. In particular, we cannot have b ⊳ I k either as any relation of I is also a relation of I ′ . Similarly, we cannot have a ⊳ I k as this would imply a ⊳ I ′ k, contradicting k ⊳ I ′ a. Proof. Because (a, b) is a Tamari inversion of I, we have b ⋪ I a and a ⋪ I b which means the relation (b, a) can be added to I as a poset. We need to check that the result I ′ is still a interval-poset.
Let us first prove that for all k such that a < k < b, we have k ⊳ I a. Let us suppose by contradiction that there exist a < k < b with k ⋪ I a and let us take the minimal k possible. Note that (a, k) is smaller than (a, b) in the lexicographic order which implies that (a, k) is not a Tamari inversion. If there is k ′ such that a < k ′ ≤ k with a ⊳ I k ′ then (a, b) is not a Tamari inversion. So there is k ′ with a ≤ k ′ < k with k ⊳ I k ′ . But because we took k minimal, we get k ′ I a which implies k ⊳ I a and contradicts the fact that (a, b) is a Tamari inversion. Now, we show that the Tamari axiom is satisfied by all triplets a ′ < k < b ′ . By Remark 2.6, we only have to consider decreasing relations. More precisely, the only cases to check are the ones where b ′ ⋪ I a ′ and b ′ ⊳ I ′ a ′ which means a I a ′ and b ′ I b (the relation is either directly added through (b, a) or obtained by transitivity). Let us choose such a couple (a ′ , b ′ ). • transitivity and contradicts our initial hypothesis.
• If a < a ′ < b then we have a ⊳ a ′ and (a, b) is not a Tamari inversion.
• The only case left is a ′ ≤ a < b ≤ b ′ . Now for k such that a ′ < k < b ′ , if k < a we get k ⊳ I a ′ by the Tamari axiom on (a, k, a ′ ). If a < k < b, we have proved that k ⊳ I a and so k ⊳ I a ′ . If b < k < b ′ , the Tamari axiom on (b, k, b ′ ) gives us k ⊳ I b and by transitivity k ⊳ I ′ a ′ . In all cases, the Tamari axiom is satisfied in I ′ for (a ′ , k, b ′ ). There is left to prove that the number of Tamari inversions of I ′ has been reduced by exactly one. More precisely: all Tamari inversions of I are still Tamari inversions of Suppose that we have b ′ ⊳ I ′ k which means that it has been added by transitivity and so we have b ′ ⊳ I b and a ⊳ I k.
No increasing relation has been created in I ′ and so a ′ ⋪ I ′ k.
we have a ⋪ I k and by the same argument as earlier that no increasing relation has been created in I ′ , a ⋪ I ′ k.
Proof of Proposition 3.9. Let I be an interval-poset containing v Tamari inversions and whose bounds are given by two binary trees [T 1 , T 2 ]. Suppose there is a chain of length k between T 1 and T 2 . In other words, we have k + 1 binary trees which connects T 1 and T 2 in the Tamari lattice. Let us look at the intervals [P i , T 2 ]. Lemma 3.10 tells us that each of them can be obtained by adding decreasing relations (b, a) to I where (a, b) ∈ TInv(I). We now apply Proposition 2.15. In our situation, it means that, for 1 ≤ j ≤ k + 1, the interval-poset of [P j , T 2 ] is an extension of every intervalposets [P i , T 2 ] with 1 ≤ i ≤ j: the Tamari inversions that were added as decreasing relations in [P i , T 2 ] are kept in [P j , T 2 ]. In other words, to obtain P i+1 from P i , one or more Tamari inversions of I are added to P i as decreasing relations. At least one Tamari inversion is added at each step which implies that v ≥ k. This is true for all chain and thus v ≥ d(I). Now, let us explicitly construct a chain between T 1 and T 2 of length v. This will give us that v ≤ d(I) and conclude the proof. We proceed inductively.
• If v = 0, then d(I) ≤ v is also 0 which means T 1 = T 2 : this is a chain of size 0 between T 1 and T 2 . • We suppose v > 0 and we apply Lemma 3.11. We take the first Tamari inversion of TInv(I) and add it to I as a decreasing relation. We obtain an interval-poset I ′ with v − 1 Tamari inversions which is a decreasing-extension of I. Then by Proposition 2.15, the bounds of I ′ are given by [T ′ 1 , T 2 ] with T ′ 1 > T 1 . By induction, we construct a chain of size v − 1 between T ′ 1 and T 2 which gives us a chain of size v between T 1 and T 2 .
The interpretation of the distance of an interval as a direct statistic on interval-posets is very useful for our purpose here as it gives an explicit way to compute it and its behavior through our involutions will be easy to state and prove. It is also interesting in itself. Indeed, this statistic appears in other conjectures on Tamari intervals, for example Conjecture 19 of [PR12] which is related to the well known open q-t-Catalan problems.

4.1.
Grafting of interval-posets. In this section, we revisit some major results of [CP15] which we will be used to define some new involutions.
Definition 4.1. Let I 1 and I 2 be two interval-posets, we define a left grafting operation and a right grafting operation depending on a parameter r. Let α and ω be respectively the label of minimal value of I 2 (shifted by the size of I 1 ) and the label of maximal value of I 1 . Let c = c 0 (I 2 ) and y 1 , . . . y c be the decreasing roots of I 2 (shifted by the size of I 1 ).
The left grafting of I 1 over I 2 with size(I 2 ) > 0 is written I 1 • I 2 . It is defined by the shifted concatenation of I 1 and I 2 along with relations y ⊳ α for all y ∈ I 1 .
The right grafting of I 2 over I 1 with size(I 1 ) > 0 is written I 1 ← − δ r I 2 with 0 ≤ r ≤ c. It is defined by the shifted concatenation of I 1 and I 2 along with relations y i ⊳ ω for 1 ≤ i ≤ c.  Figure 15 gives an example. Note that the vertices of I 2 are always shifted by the size of I 1 . For simplicity, we do not always recall this shifting: when we mention a vertex x of I 2 in a grafting, we mean the shifted version of x. These two operations were defined in [CP15, Def. 3.5]. Originally, the right grafting was defined as a single operation ← − δ which result was a formal sum of interval-posets. In this paper, it is more convenient to cut it into different sub-operations depending on a parameter. We can use these operations to uniquely decompose interval-posets: this will be explained in Section 4.2. First, we will study how the different statistics we have defined are affected by the operations. We start with the contact vector C which is equal to the final forest vector DC.
Proof. First, remember that, by Proposition 3.6, contacts can be directly computed on the final forest of the interval-posets: the non-final contacts correspond to the number of components and c v for 1 ≤ v ≤ n is the number of decreasing children of the vertex v. Now, in the left grafting I 1 • I 2 , the two final forests are simply concatenated. In particular, c 0 (I 1 • I 2 ) = c 0 (I 1 ) + c 0 (I 2 ). The contact vector C(I 1 • I 2 ) is then formed by this initial value followed by the truncated contact vector of I 1 , then an extra 0 which correspond to c n , then the truncated contact vector of I 2 .
The contacts of the right grafting I 1 ← − δ i I 2 depend on the parameter i. Indeed, each added decreasing relation merges one component of the final forest of I 2 with the last component of the final forest of I 1 and thus reduces the number of components by one. As a consequence, we have c 0 (I 1 ← − δ i I 2 ) = c 0 (I 1 ) + c 0 (I 2 ) − i. The contact vector is formed by this initial value followed by the truncated contact vector of I 1 , then the new number of decreasing children of n which is i by definition, then the truncated contact vector of I 2 .
Proof. When we compute I 1 • I 2 , we add increasing relations from all vertices of I 1 to the first vertex α of the shifted copy of I 2 . In other words, we attach all increasing roots of I 1 to a new root α. The number of components in the initial forest of I 1 • I 2 is then given by ic ∞ (I 2 ) (the last component contains I 1 ) and the number of increasing children of α is given by ic ∞ (I 1 ). Other number of increasing children are left unchanged and we thus obtain the expected vector.
In the computation of I 1 ← − δ i I 2 , the value of i only impacts the decreasing relations and thus does not affects the vector IC. No increasing relation is added which means that the initial forests of I 1 and I 2 are only concatenated and by looking at connected components, we obtain ic ∞ (I 1 ← − δ i I 2 ) = ic ∞ (I 1 ) + ic ∞ (I 2 ). The vector IC is formed by this initial value followed by the truncated initial forest vector of I 2 , then an extra 0 which correspond to ic 1 (I 2 ), then the truncated initial forest vector of I 1 .
To understand how the rise vector behaves through the grafting operations, we first need to interpret the grafting on the upper bound Dyck path of the interval. We start with the left grafting. • r 0 (I 1 • I 2 ) = r 0 (I 1 ); • R(I 1 • I 2 ) = (r 0 (I 1 ), r 1 (I 1 ), . . . , r n−1 (I 1 ), r 0 (I 2 ), r 1 (I 2 ), . . . , r m−1 (I 2 )). Proof. The definition of I 1 • I 2 states that we add all relations (i, α) with i ∈ I 1 and α the first vertex of I 2 . This is the same as adding all relations (i, α) where i is an increasing root of I 1 (the other relations are obtained by transitivity). The increasing roots of I 1 correspond to the up-steps of D 1 which corresponding down-steps do not have a following up-step, i.e., the up-steps corresponding to final down-steps of D 1 . By concatenating D 1 and D 2 , the first up-step of D 2 is now the first following up-step of the final down-steps of D 1 : this indeed adds the relations from the increasing roots of I 1 to the first vertex of I 2 . The expressions for the initial rise and rise vectors follow immediately by definition.
The effect of the right grafting on the rise vector is a bit more technical. For simplicity, we only study the case where I 1 is of size one which is the only one we will use in practice. Proof. The right-grafting only adds decreasing relations. On the initial forests, it is then nothing but a concatenation of the two initial forests. In particular, in the case of u ← − δ i I, no increasing relation is added from the vertex one to any vertex of I. On the upper bound Dyck path, this means that the down-step corresponding to the initial up-step is not followed by any up-steps: the Dyck path of I has to be nested into this initial up-step. The expressions for the rise vector follows immediately.
Remark 4.6. When applying a right-grafting on u, the interval-poset of size 1, the rise vector and the initial forest vector have similar expressions: This will be a fundamental property when we define our involutions. Note also that we can have size(I 2 ) = 0 in all these expressions. Now, the only statistic which is left to study through the grafting operations is the distance. Recall that by Proposition 3.9, it is given by the number of Tamari inversions. In the same way as for the R vector, it is more complicated to study on the right grafting in which case, we will restrict ourselves to size(I 1 ) = 1.
Proposition 4.7. Let I 1 and I 2 be two interval-posets, and u be the interval-poset of size one. Then Look for example at Figure 16: the Tamari inversion (1, 3) of I 1 and (1, 2) of I 2 are kept through I 1 • I 2 and no other Tamari inversion is added. For the right grafting, you can look at Figure 17: the intervalposet I 2 only has one Tamari inversion (1, 3) and we have c 0 (I 2 ) = 2. You can check that d(u ← − δ 1 I 2 ) = 2 = 1+2−1, the two Tamari inversions being (2, 4) and (1, 4).
Proof. We first prove d(I 1 • I 2 ) = d(I 1 ) + d(I 2 ). The condition for a couple (a, b) to be a Tamari inversion is local: it depends only on the values a ≤ k ≤ b. Thus, because the local structure of I 1 and I 2 is left unchanged, any Tamari inversion of I 1 and I 2 is kept in I 1 • I 2 . Now, suppose that a ∈ I 1 and b ∈ I 2 . Let α be the label of minimal value in I 2 (which has been shifted by the size of I 1 ). By definition, we have a < α ≤ b and a ⊳ α in I 1 • I 2 : (a, b) is not a Tamari inversion. Now, let I = u ← − δ i I 2 with 0 ≤ i ≤ c 0 (I 2 ) and let us prove that d(I) = d(I 2 ) + c 0 (I 2 ) − i. Once again, note that the Tamari inversions of I 2 are kept through the right grafting. For the same reason, the only Tamari inversions that could be added are of the form (1, b) with b ∈ I 2 . Now, let b be a vertex of I 2 which is not a decreasing root. This means there is a < b with b ⊳ I 2 a. In I, the interval-poset I 2 has been shifted by one and so we have: 1 < a < b with b ⊳ I a: (1, b) is not a Tamari inversion of I. Let b be a decreasing root of I 2 . If b ⊳ I 1 then (1, b) is not a Tamari inversion. If b ⋪ I 1, we have that: by construction, there is no a ∈ I 2 with 1 ⊳ I a; because b is a decreasing root there is no a ∈ I 2 with a < b and b ⊳ a. • The operation . In practice, we think of it as I L • (u ← − δ r I R ).
The parameter r identifies which element is I in the composition sum.
Definition 4.10. Let T be a binary tree of size n. We write v 1 , . . . , v n the nodes of T taken in in-order (following the binary search tree labeling). Let ℓ : {v 1 , . . . , v n } → N be a labeling function on T . For all subtrees T ′ of T , we write size(T ′ ) the size of the subtree and labels(T ′ ) := v i ∈T ′ ℓ(v i ) the sum of the labels of its nodes.
We say that (T, ℓ) is a Tamari interval grafting tree, or simply grafting tree if the labeling ℓ satisfies that for every node v i , we have An example is given in Figure 19: the vertices v 1 , . . . , v 8 are written in red above the nodes, whereas the labeling ℓ is given inside the node. For example, you can check the rule on the root v 4 , we have size(T R (v 4 )) − labels(T R (v 4 )) = 4 − 1 = 3 and indeed ℓ(v 4 ) = 2 ≤ 3.
The rule is satisfied on all nodes. Note that if the right subtree of a node is empty (which is the case for v 1 , v 3 , v 6 , and v 8 ) then the label is always 0.  Proof. First, let us check that we can obtain a interval-poset from a grafting tree. We read the grafting tree as an expression tree where each empty subtree is replaced by an entry as an empty interval-poset and each node corresponds to the operation I L • u ← − δ r I R where r is the label of the node, I L and I R the respective results of the left and right subtrees, and u the interval poset of size 1. In other words, the interval-poset I = ∆ −1 (T, ℓ) where (T, ℓ) is a grafting tree is computed recursively by • if T is empty then I = ∅; and ℓ L and ℓ R the labeling function ℓ restricted to respectively T L and T R .
We need to check that the operation u ← − δ r ∆ −1 (T R , ℓ R ) is well defined i.e, in the case where T is not empty, that we have 0 ≤ r ≤ c 0 (∆ −1 (T R , ℓ R )). We do that by also proving by induction that c 0 (∆ −1 (T, ℓ)) = size(T ) − labels(T ). This is true in the initial case where T is empty: c 0 (∅) = 0. Now, suppose that T = v k (T L , T R ) with ℓ(v k ) = r and that the property is satisfied on (T L , ℓ L ) and (T R , ℓ R ). We write I L = ∆ −1 (T L , ℓ L ) and I R = ∆ −1 (T R , ℓ R ). In this case, I ′ := u ← − δ r I R is well defined because we have by definition that r ≤ size(T R ) − labels(T R ), which by induction is c 0 (I R ). Besides, by Proposition 4.2, we have c 0 (I ′ ) = 1 + c 0 (I R ) − r. We now compute I = I L • I ′ and we get c 0 (I) = c 0 (I L ) + 1 + c 0 (I R ) − r which is by induction size(T L ) − labels(T L ) + 1 + size(T R ) − labels(T R ) − r = size(T ) − labels(T ).
Conversely, it is clear from Proposition 4.8 that the grafting decomposition of an interval-poset I gives a unique labeled binary tree. We need to prove that the condition on the labels holds. Once again, this is done inductively. An empty interval-poset gives an empty tree and the condition holds. Now if I decomposes into the triplet (I L , I R , r) we suppose that the condition holds on (T L , ℓ L ) = ∆(I L ) and (T R , ℓ R ) = ∆(I R ). We know that 0 ≤ r ≤ c 0 (I R ) and we have just proved that c 0 (I R ) is indeed size(T R ) − labels(T 2 ). In other words, the grafting tree of an interval-poset can be obtained directly without using the recursive decomposition. Also, the tree T only depends on the initial forest and the labeling ℓ only depends on the final forest.
Proof. We prove the result by induction on I. If I is empty, there is nothing to prove. We then suppose that I decomposes into a triplet (I L , I R , r) with k = size(I L ) + 1. We suppose by induction that the proposition is true on I L and For example, on Figure 19, we have all d i = 0 except for d 4 = 4 − 1 − 2 = 1 and d 5 = 3 − 1 = 2. This indeed is consistent with d(I) = 3, the 3 Tamari inversions being (4, 7), (5, 6), and (5, 7). More precisely, the number d i is the number of Tamari inversions of the form (i, * ).
Proof. Once again, we prove the property inductively. This is true for an empty tree where we have d(I) = 0. Now, let I be a nonempty interval-poset, then I decomposes into a triplet (I L , I R , r) with Proposition 4.7 gives us (4.12) 4.3. Left branch involution on the grafting tree. We now give an interesting involution on the grafting tree which in turns gives an involution on Tamari intervals. We call right hanging binary trees the binary trees whose left subtree is empty. An alternative way to see a binary tree is to understand it as list of right hanging binary trees grafted together on its left-most branch. For example, the tree of Figure 19 can be decomposed into 3 right hanging binary trees : the one with vertex v 1 , the one with vertices v 2 and v 3 and the one with vertices v 4 to v 8 .   Figure 20, The only thing to check is that ℓ ′ still satisfies the grafting tree condition. This is immediate. Indeed, for v i ∈ T , and T R (v i ) its right subtree, we have ℓ( its right subtree, even though T ′ R might be different from T R , the statistics are preserved: size(T ′ R (v i ′ )) = size(T R (v i )) and labels(T ′ R (v i ′ )) = labels(T R (v i )), because the involution only acts on left branches.
As a consequence, we now have an involution on Tamari intervals.

Definition 4.17 (The Left Branch Involution). The left branch involution on Tamari intervals is defined by the left branch involution on their grafting trees.
φ(I) := ∆ −1 (φ(∆(I))) (4.13) The grafting tree seems to be the most natural object to describe the involution. Indeed, even though it can be easily computed on intervalposets using decomposition and graftings, we have not seen any simple direct description of it. Furthermore, if we understand the interval as a couple of a lower bound and upper bound, then the action on the upper bound is simple: the shape of the upper bound binary tree is given by the grafting tree and so the involution on the upper bound is only the classical left-branch involution. Nevertheless, the action on the lower bound cannot be described as an involution on binary trees: it depends on the corresponding upper bound. One way to understand this involution is that we apply the left-branch involution on the upper bound binary tree and the lower bounds "follows" in the sense given by the labels of the grafting tree.  In other words, the involution exchanges the rise vector and initial forest vector while leaving unchanged the number of contacts, the contact monomial, and the distance.
Proof. Points (4.14) and (4.15) are immediate. Indeed, (4.5) tells us that c 0 (I) is given by size(∆(I)) − labels(∆(I)) : this statistic is not changed by the involution. Now remember that, by Proposition 4.2, the values c 1 (I), . . . , c n (I) are given by ℓ(v 1 ), . . . , ℓ(v n ), so . This monomial is commutative and the involution sending ℓ to ℓ ′ only applies a permutation on the indices: the monomial itself is not changed. Also, we always have ℓ(v n ) = ℓ ′ (v n ) = 0 so the division by x 0 is still possible after the permutation and still removes the last value x ℓ ′ (vn) . As an example, on Figure 20, we have C( Point (4.16) is also immediate by Proposition 4.14. Indeed, for all We prove point (4.17) by induction. It is trivially true when size(I) = 0 (both vectors are empty). Now suppose that I is an interval-poset of size n > 0. Let (T, ℓ) = ∆(I), then T is a non-empty binary tree which can be seen as a list of k non-empty right hanging binary trees T 1 , . . . , T k where T 1 is the left-most one and T k is at the root. We write I 1 , . . . I k the corresponding interval-posets and I ′ 1 , . . . , I ′ k their respective images through φ. By definition of ∆(I), we have that At this point, either k > 1 which means the sizes of each intervals I 1 , . . . , I k is strictly smaller than n and we can conclude by induction. Otherwise, k = 1 and φ(I) is a right hanging binary tree. We then have where u is the interval-poset of size 1, J is an interval-poset of size n − 1 and r is a parameter 0 ≤ r ≤ c 0 (J). We write J ′ = φ(J). By

The complement involution and rise-contact involution.
The left-branch involution φ is not yet what we need to prove Theorem 3.4. Indeed, we want an involution which exchanges the contact monomial with the rise monomial and the number of non-final contacts with the number of initial rises. Nevertheless, the left-branch involution will be our crucial element to build it, combined with another involution which we define now.   Proof. Every increasing relation a ⊳ I b is sent to a decreasing relation (n+1−a) ⊳ ψ(I) (n+1−b). In particular, each connected component of the initial forest of I is sent to exactly one connected component of the final forest of ψ(I) and so ic ∞ (I) = dc 0 (ψ(I)). Now, if a vertex b has k increasing children in I, its image (n+1−b) has k decreasing children in ψ(I) so ic b (I) = dc n+1−b (ψ(I)). Remember that IC * reads the numbers of increasing children in reverse order from n to 2 whereas DC * reads the number of decreasing children in the natural order from 1 = n+1−n to n − 1 = n + 1 − 2. We conclude that IC(I) = DC(ψ(I)).
More precisely, (a, b) is a Tamari inversion of I if and only if (n + 1 − b, n + 1 − a) is a Tamari inversion of ψ(I).
Proof. Let a < b be two vertices of I, we set a ′ = n + 1 − b and b ′ = n + 1 − a.
• There is a ≤ k < b with b ⊳ I k if and only if there is In other words, (a, b) is a Tamari inversion of I if and only if (a ′ , b ′ ) is a Tamari inversion of ψ(I). By Proposition 3.9, this gives us d(I) = d(ψ(I)).
You can check on Figure 21 that I has 3 Tamari inversions (1, 5), (2, 3), and (2, 5) which give respectively the Tamari inversions (4, 8), (6, 7), and (4, 7) in ψ(I). We are now able to state the following Theorem which gives an explicit combinatorial proof of Theorem 3.4. We give an example computation on Figure 22. You can run more examples and compute tables for all intervals using the provided live Sage-Jupyter notebook [Pon].

5.
The m-Tamari case 5.1. Definition and statement of the generalized result. The m-Tamari lattices are a generalization of the Tamari lattice where objects have a (m + 1)-ary structure instead of binary. They were introduced in [BPR12] and can be described in terms of m-ballot paths. A m-ballot path is a lattice path from (0, 0) to (nm, n) made from horizontal steps (1, 0) and vertical steps (0, 1) which always stays above the line y = x m . When m = 1, a m-ballot path is just a Dyck path where up-steps and down-steps have been replaced by respectively vertical steps and horizontal steps. They are well known combinatorial objects counted by the m-Catalan numbers (5.1) 1 mn + 1 (m + 1)n n .
They can also be interpreted as words on a binary alphabet and the notion of primitive path still holds. Indeed, a primitive path is a mballot path which does not touch the line y = x m outside its extremal points. From this, the definition of the rotation on Dyck path given in Section 2.2 can be naturally extended to m-ballot-paths, see Figure 23. When interpreted as a cover relation, the rotation on m-ballot paths induces a well defined order, and even a lattice [BPR12]. This is what we call the m-Tamari lattice or T (m) n , see Figure 24 for an example. The intervals of m-Tamari lattices have also been studied. In [BMFPR11], it was proved that they are counted by (5.2) I n,m = m + 1 n(mn + 1) (m + 1) 2 n + m n − 1 .
They were also studied in [CP15] where it was shown that they are in bijection with some specific families of Tamari interval-posets. Our goal here is to use this characterization to generalize Theorem 3.4 to intervals of m-Tamari, thus proving Conjecture 17 of [PR12]. First, let us introduce the m-statistics which correspond to the classical cases statistics defined in Definition 3.1.
Definition 5.1. Let B be a m-ballot path. We define the following m-statistics. An example is given on Figure 25. When m = 1, this is the same as Definition 3.1. Note also that we will later define a bijection between m-ballot paths and certain families of Dyck paths which also extends to intervals: basically any element of T (m) n can also be seen as an element of T n×m but the statistics are not exactly preserved which is why we use slightly different notations for m-statistics to avoid any confusion. . . , a n ], the area vector of B. We partition the values of a such that a i and a j are in the same set if a i = a j and for all i ′ such that i ≤ i ′ ≤ j, then a i ′ ≥ a i . Let λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ k ) be the integer partition obtained by keeping only the set sizes and let e(B, X) = x λ 1 . . . x λ k a monomial on a commutative alphabet X. Then e(B, X) = C m (B, X) with x 0 = 1.
The definition of e(B, X) comes from [PR12, Conjecture 17]. As an example, the area vector of the path from Figure 25 is (0, 1, 2, 4, 2, 4, 4, 0). The set partition is {{a 1 , a 8 }, {a 2 }, {a 3 , a 5 }, {a 4 }, {a 6 , a 7 }}. In particular, the area vector always starts with a 0 and each new 0 corresponds to a contact between the path and the line. Here, we get λ = (2, 2, 2, 1, 1) which indeed gives e(B, X) = x 2 1 x 3 2 = C(B, X) at x 0 = 1. Proof. If the step u i starts at a point (x, y), then we have by definition my = x + a i . In particular, if a i = a j , then u i and u j both have a contact with a same affine line s of slope 1 m . Then a i and a j belong to the same set in the partition if and only if the path between u i and u j stays above the line s. More precisely, the line s cuts a section p of the path, starting at some point (a, b + j m ) where (a, b) is the starting point of a vertical step and 1 ≤ j ≤ m. The non-final contacts of this path p with the line s are exactly the vertical steps u k with a k = a i . The final contact corresponds either to the end of the path B or to a horizontal step: it does not correspond to an area a k = a i .
As for the classical case, we now extend those definitions to intervals of the m-Tamari lattice. Finally, the definition of distance naturally extends to m-Tamari. We can now state the generalized version of Theorem 3.4.
As the m-Tamari lattice can be understood as an upper ideal of the Tamari lattice, it follows that the intervals of T (m) n are actually a certain subset of intervals of T n×m : they are the intervals where both upper and lower bounds are m-Dyck paths (in practice, it is sufficient to check that the lower bound is a m-Dyck path). It is then possible to represent them as interval-posets. This was done in [CP15] where the following characterization was given.
for all 1 ≤ i ≤ n.  Figure 27. The m-binary trees have a (m + 1)-ary recursive structure: this is the key element to prove Theorem 5.9 and we will also use it in this paper.
Definition 5.10. The m-binary trees are defined recursively by being either the empty binary tree or a binary tree T of size m×n constructed from m + 1 subtrees T L , T R 1 , . . . , T Rm such that • the sum of the sizes of T L , T R 1 , . . . , T Rm is mn − m; • each subtree T L , T R 1 , . . . , T Rm is itself a m-binary tree; • and T follows the structure bellow.

T Rm
The left subtree of T is T L . The right subtree of T is constructed from T R 1 , . . . , T Rm by the following process: graft a an extra node to the left of the leftmost node of T R 1 , then graft T R 2 to the right of this node, then graft an extra node to the left of the leftmost node of T R 2 , then graft T R 3 to the right of this node, and so on.
Note that in total, m extra nodes were added: we call them the mroots of T . Figure 28 gives two examples of m-binary trees for m = 2 with their decompositions into 3 subtrees. More examples and details about the structure can be found in [CP15]. In particular, m-binary trees are the images of m-Dyck paths through the bijection of Definition 2.11.
When working on the classical case, we could safely identify an interval of the Tamari lattice and its representing interval-poset. For m = 1, we need to be a bit more careful and clearly separate the two notions. Indeed, the m-statistics from Definition 5.3 of an interval of T (m) n are not equal to the statistics of its corresponding interval-poset Figure 28. Examples of m-binary trees for m = 2: T L is in red, T R 1 is in dotted blue and T R 2 is in dashed green. In the second example, T R 1 is empty.
from Definition 3.2. They can anyway be retrieved through simple operations.
Proposition 5.11. Let I be an interval of T Proof. All identities related to rises and contacts are a direct consequence of Proposition 5.7. Only (5.14) needs to be proved, which is actually also direct: T (m) n is isomorphic to the ideal of m-Dyck path in T n×m and so the distance between two paths in the lattice stays the same. In particular, m-interval-posets are rise-m-divisible but not necessary contact-m-divisible. Besides, we saw that rise-m-divisible Dyck paths were exactly m-Dyck paths, but the set of rise-m-divisible intervalposets is not equal to m-interval-posets. Indeed, an interval whose upper bound is a m-Dyck path is rise-m-divisible but it can have a lower bound which is not a m-Dyck path and so it is not a m-intervalposet.
Furthermore, it is quite clear that the set of m-interval-posets is not stable through the rise-contact involution β. Indeed, the image of a m-interval-poset would be contact-m-divisible but not necessary risem-divisible. In this section, we describe a bijection between the set of m-interval-posets and the set of rise-contact-m-divisible intervals. This bijection will allows us to define an involution on m-interval-posets which proves Theorem 5.5.
Definition 5.13. Let (T, ℓ) be a grafting tree of size nm and v 1 , . . . , v nm be the nodes of T taken in in-order. We say that (T, ℓ) is a m-graftingtree if ℓ(v i ) ≥ 1 for all i such that i ≡ 0 mod m.
Proposition 5.14. An interval-poset I is a m-interval-poset if and only if ∆(I) is a m-grafting-tree.
As an example, the top and bottom grafting trees of Figure 30 are m-grafting trees: you can check that every odd node has a non-zero label. The corresponding m-interval-posets are drawn on the same lines. Proposition 5.14 is a direct consequence of Theorem 5.9 and Proposition 4.12.
Proof. This is immediate by Proposition 4.12: (T, ℓ) corresponds to a m-interval-poset I. In particular, the upper bound of I is a m-binary tree which is equal to T .
Proof. This proposition contains different results which we organize as claims and prove separately.
These two properties are intrinsically linked, we will prove both at the same time by induction on the recursive structure of m-binarytrees. Let (T, ℓ) be a m-grafting tree. By Proposition 5.15, T is a m-binary tree. If T is empty, then there is nothing to prove. Let us suppose that T is non empty: it can be decomposed into m + 1 subtrees T L , T R 1 , . . . , T Rm which are all m-grafting trees. By induction, we suppose that they satisfy the claim.
Let us first focus on the case where T L is the empty tree. Then v 1 (the first node in in-order) is the root and moreover, the m-roots are v 1 , . . . v m . We call T 1 , T 2 , . . . , T m the subtrees of T whose roots are respectively v 1 , . . . , v m (in particular, T 1 = T ). See Figure 29 for an illustration. Figure 29. Illustration of T 1 , . . . , T m In particular, for 1 ≤ k < m, the tree T k follows a structure that depends on T R k and T k+1 as shown in Figure 29 and T m depends only on T Rm . Note that T 2 , . . . T k are grafting trees but they are not mgrafting trees whereas T R 1 , . . . , T Rm are. Following Definition 4.10, the structure gives us for 1 ≤ k < m and (5.17) ℓ(v m ) ≤ c 0 (T Rm , ℓ).
Also, for 1 ≤ k < m, we have ℓ ′ (v k ) = m(ℓ(v k ) − 1) ≥ 0 (indeed remember that ℓ(v k ) ≥ 1 because (T, ℓ) is a m-grafting-tree) and ℓ ′ (v m ) = mℓ(v m ) ≥ 0. To prove that (T, ℓ ′ ) is a grafting tree, we need to show We simultaneously prove The case k = 1 in (5.18) and (5.20) proves the claim. We start with k = m and then do an induction on k decreasing down to 1. By hypothesis, we know that (T Rm , ℓ) satisfies the claim. In particular (T Rm , ℓ ′ ) is a grafting tree and c 0 (T Rm , ℓ ′ ) = m c 0 (T Rm , ℓ). i.e., case k = m of (5.20). Now, we choose 1 ≤ i < m and assume (5.18) and (5.20) to be true for k > i. We have ℓ ′ (v i ) = m (ℓ(v i ) − 1), so (5.16) gives us ℓ ′ (v i ) ≤ m c 0 (T R i , ℓ) + m c 0 (T i+1 , ℓ) − m (5.23) = c 0 (T R i , ℓ ′ ) + c 0 (T i+1 , ℓ ′ ) + i − m using (5.20) with k = i + 1. As i < m, this proves (5.18) for k = i. Now, the structure of T i gives us The case where T L is not the empty tree is left to consider but actually follows directly. The claim is true on T L by induction as its size is strictly smaller than T . LetT be the tree T where you remove the left subtree T L . ThenT is still a m-grafting tree and the above proof applies. The expansion on T consists of applying the expansion independently on T L andT and we get c 0 (T, ℓ ′ ) = c 0 (T L , ℓ ′ )+c 0 (T , ℓ ′ ) = m c 0 (T, ℓ).
T is still a m-binary tree, which by Proposition 4.12, means that the upper bound of ∆ −1 (T, ℓ ′ ) is a m-binary tree: it corresponds to a m-Dyck path and is then m-rise-divisible. We have just proved that c 0 (T, ℓ ′ ) = m c 0 (T, ℓ) is a multiple of m. By Proposition 4.2 the rest of the contact vector is given by reading the labels on T : by definition of ℓ ′ , all labels are multiples of m.
We define (T, ℓ) = contract(T, ℓ ′ ) to make it the inverse of the expand operation: As earlier, we simultaneously prove that (T, ℓ) is a m-grafting tree and that c 0 (T, ℓ) = c 0 (T,ℓ ′ ) m . Our proof follows the exact same scheme as for Claim 1. We recursively decompose T into T L , T R 1 , . . . , T Rm . As earlier, the only case to consider is actually when T L is empty. We use the decomposition of T depicted in Figure 29 and prove (5.20) and (5.16) by induction on k decreasing from m to 1. The case where k = m is straightforward: we have that (5.19) implies (5.17) and (5.22) is still true. Now, we choose 1 ≤ i < m and assume (5.16) and (5.20) to be true for k > i. Using (5.18), we get We have 0 < i m < 1 and because ℓ(v i ) is an integer then (5.16) is true. Besides, by definition of ℓ, ℓ(v i ) ≥ 1 which satisfies the m-grafting tree condition. The rest of the induction goes smoothly because (5.25) is still valid.
The expand and contract operations are the final crucial steps that allow us to define a the m-contact-rise involution and prove Theorem 5.5. Before that, we need a last property to understand how the distance statistic behaves through the transformation.  for 1 ≤ i ≤ n and 1 ≤ j < m. In other words, every value of the contact vector of (T, ℓ) is m times the corresponding value of the mcontact vector of I. Besides, as the expansion does not affect the initial forest ofĨ, we also have that every value of the rise vector of (T, ℓ) is m times the corresponding value in the m-rise vector of I.
The grafting tree (T, ℓ) is rise-contact-m-divisible. This property is preserved through the β involution and if (T ′ , ℓ ′ ) = β(T, ℓ), then the contact vector of (T ′ , ℓ ′ ) is a permutation of the rise vector of (T, ℓ) and vice-versa. Because (T ′ , ℓ ′ ) is rise-contact-m-divisible, we can apply the contract operation and we get a m-interval-poset which corresponds to some interval J of T (m) n . The m-rise and m-contact vectors of J are computed respectively from the rise and contact vectors of (T ′ , ℓ ′ ) by dividing all values by m. This proves (5.34) and (5.35).
For (5.36), see that the distance statistic is only affected by the expand and contract operations. The expand only applies an affine transformation which does not depend on the shape of T and is then reverted by the application of contract later on.