Three Fuss-Catalan posets in interaction and their associative algebras

We introduce $\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $\delta$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak Bruhat order of permutations, we construct algebras whose products form intervals of the lattices of $\delta$-cliff. We provide necessary and sufficient conditions on $\delta$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $\delta$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra.

The theory of combinatorial Hopf algebras takes a prominent place in algebraic combinatorics. The Malvenuto-Reutenauer algebra FQSym [MR95,DHT02] is a central object in this theory. This structure is defined on the linear span of all permutations and the product of two permutations has the notable property to form an interval of the right weak Bruhat order. Moreover, FQSym admits a lot of substructures, like the Loday-Ronco algebra of binary trees PBT [LR98,HNT05] and the algebra of noncommutative symmetric functions Sym [GKL + 95]. Each of these structures brings out in a beautiful and somewhat unexpected way the combinatorics of some partial orders, respectively the Tamari order [Tam62] and the Boolean lattice, playing the same role as the one played by the right weak Bruhat order for FQSym. To be slightly more precise, all these algebraic structures have, as common point, a product · which expresses, on their so-called fundamental bases {F } , as where is a partial order on basis elements, and and are some binary operations on basis elements (in most cases, some sorts of concatenation operations).
The point of departure of this work consists in considering a different partial order relation on permutations and ask to what extent analogues of FQSym and a similar hierarchy of algebras arise in this context. We consider here first a very natural order on permutations: the componentwise ordering on Lehmer codes of permutations [Leh60]. A study of these posets Cl 1 ( ) appears in [Den13]. Each poset Cl 1 ( ) is an order extension of the right weak Bruhat order of order . To give a concrete point of comparison, the Hasse diagrams of the right weak Bruhat order of order 3 and of Cl 1 (3) are respectively As we can observe, the right weak Bruhat order relation on permutations of size 3 is included into the order relation of Cl 1 (3).
In this work, we consider a more general version of Lehmer codes, called δ-cliffs, leading to distributive lattices Cl δ . Here δ is a parameter which is a map N \ {0} → N, called range map, assigning to each position of the words a maximal allowed value. The linear spans Cl δ of these sets are endowed with a very natural product related to the intervals of Cl δ . Some properties of this product are implied by the general shape of δ. For instance, when δ is so-called valley-free, Cl δ is an associative algebra, and when δ is weakly increasing, Cl δ is free as a unital associative algebra. The particular algebra Cl 1 is in fact isomorphic to FQSym, so that for any range map δ, Cl δ is a generalization of this latter. For instance, when δ is the map m satisfying m( ) = ( − 1) with ∈ N, then all Cl m are free associative algebras whose bases are indexed by increasing trees wherein all nodes have + 1 children.
In the same way as the Tamari order can be defined by restricting the right weak Bruhat order to some permutations, one builds three subposets of Cl δ by restricting to particular δ-cliffs. This leads to three families Av δ , Hi δ , and Ca δ of posets. When δ is the particular map m defined above with 0, the underlying sets of all these posets of order 0 are enumerated by the -th -Fuss-Catalan number [DM47] cat ( ) := 1 + 1 These posets have some close interactions: when δ is an increasing map, Hi δ is an order extension of Ca δ , which is itself an order extension of Av δ . Besides, Hi m (resp. Ca m ) generalizes for any 0 the Stanley lattice [Sta75,Knu04] (resp. Tamari lattice), which occurs when m = 1. Our generalization of Tamari lattices is different from the classical one introduced in [BPR12]. Besides, from these posets Hi m and Ca m , one defines respectively two quotient algebras Hi and Ca of Cl m . Notably, The algebra Ca 1 is isomorphic to PBT, and the other ones Ca , 2, are not free as associative algebras.
This paper is organized as follows.
Section 1 is intended to introduce δ-cliffs and to set some notations and recalls about poset theory. As a by-product in the process of establishing links between the posets Cl δ ( ) and the weak Bruhat order, we introduce an alternative poset (Cl δ ( ) ′ ) when δ satisfies some particular conditions. We prove that when δ = 1, the obtained poset is the weak Bruhat order and we conjecture that for all authorized range maps δ, the posets (Cl δ ( ) ′ ) are semi-distributive lattices. Besides, even if the posets Cl δ ( ) have a very simple structure, they contain interesting subposets ( ). To study these substructures, we establish a series of sufficient conditions on ( ) for the fact that these posets are EL-shellable [BW96,BW97], are lattices (and give algorithms to compute the meet and the join of two elements), and are constructible by interval doubling [Day79]. Moreover, under some precise conditions, each subposet ( ) can be seen as a geometric object in R . We call this the geometric realization of ( ). We introduce here the notion of cell and expose a way to compute the volume of the geometrical object.
Next, in Section 2, we study the posets Av δ , Hi δ , and Ca δ . For each of these, we provide some general properties (EL-shellability, lattice property, constructibility by interval doubling), and describe its input-wings, output-wings, and butterflies elements, that are elements having respectively a maximal number of covered elements, covering elements, or both properties at the same time. We observe a surprising phenomenon: some posets Av δ , Hi δ , or Ca δ are isomorphic to their subposets restrained on input-wings, output-wings, or butterflies elements. Moreover, a notable link among other ones is that the subposet of Ca m ( ) is isomorphic to the subposet of Hi m−1 ( ) restrained to its input-wings. We also study further interactions between our three families of Fuss-Catalan posets: there are for instance bijective posets morphisms (but not poset isomorphisms) between Av δ and Ca δ , and between Ca δ and Hi δ , when δ is increasing.
Finally, Section 3 presents a study of the algebra Cl δ . We start by introducing a natural coproduct on Cl δ in order to obtain by duality a product, associative in some cases. Three alternative bases of Cl δ are introduced, including two that are multiplicative and are defined from the order on δ-cliffs. We then rely on these bases to give a presentation by generators and relations of Cl δ . When δ is valley-free and is 1-dominated (that is a certain condition on range maps), Cl δ admits a finite presentation (a finite number of generators and a finite number of nontrivial relations between the generators). When δ is weakly increasing, Cl δ is free as an associative algebra. We end this work by constructing, given a subfamilly of Cl δ , a quotient space Cl of Cl δ isomorphic to the linear span of . A sufficient condition on to have moreover a quotient algebra of Cl δ is introduced. We also describe a sufficient condition on for the fact that the product of two basis elements of Cl is an interval of a poset ( ). These results are applied to construct and study the two quotients Hi := Cl Hi m and Ca := Cl Ca m of Cl m . The algebra Ca 1 is isomorphic to the Loday-Ronco algebra and the other algebras Ca , 2, provide generalizations of this later which are not free. On the other hand, for any 1, all Hi are other associative algebras whose dimensions are also Fuss-Catalan numbers and are not free. This paper is an extended version of [CG20] containing the proofs of the presented results and presenting new ones as the geometrical aspects of the studied posets.  Given a range map δ, a word of nonnegative integers of length is a δ-cliff if for any ∈ [ ], ∈ δ( )]. The size | | of a δ-cliff is its length as a word, and the weight ω( ) of is the sum of its letters. The graded set of all δ-cliffs where the degree of a δ-cliff is its size, is denoted by Cl δ . In the sequel, for any 0, we shall denote by m the range map satisfying Last sequence is Sequence A001147 of [Slo].

Lehmer codes and permutations.
There is a classical correspondence between permutations and Lehmer codes [Leh60], that are certain words of integers. Here, we consider a slight variation of Lehmer codes, establishing a bijection between the set of 1-cliffs of size and the set of permutations of the same size. Given a permutation σ of size , let the 1-cliff such that for any ∈ [ ], is the number of values such that < while σ −1 ( ) > σ −1 ( ). We denote by leh(σ ) the 1-cliff thus associated with the permutation σ . For instance, leh(436512) = 002323.
1.1.3. Weakly increasing range maps and increasing trees. Given a rooted weakly increasing range map δ, let ∆ δ : N \ {0} → N be the map defined by ∆ δ ( ) := δ( + 1) − δ( ). For instance, for any 0, ∆ ω = 0 ω , and for any 0, ∆ m = ω . A δ-increasing tree is a planar rooted tree where internal nodes are bijectively labeled from 1 to , any internal node labeled by ∈ [ ] has arity ∆ δ ( ) + 1, and every children of any node labeled by ∈ [ ] are leaves or are internal nodes labeled by ∈ [ ] such that > . The size of such a tree is its number of internal nodes. The leaves of a δ-increasing tree are implicitly numbered from 1 to its total number of leaves from left to right.
Observe that, regardless of any particular condition on δ, any δ-cliff of size 1 recursively decomposes as = ′ where ∈ δ( )] and ′ is a δ-cliff of size − 1. Relying on this observation, when δ is rooted and weakly increasing, let tree δ be the map sending any δ-cliff of size to the δ-increasing tree of size recursively defined as follows. If = 0, tree δ ( ) is the leaf. Otherwise, by using the above decomposition of , tree δ ( ) is the tree obtained by grafting on the ( + 1)-st leaf of the tree tree( ′ ) a node of arity ∆ δ ( ) + 1 labeled by . For instance, tree 2 (0230228) = Proof. Let us first prove that tree δ is a well-defined map. This can be done by induction on and arises from the fact that, for any ∈ Cl δ ( ), the total number of leaves of tree δ ( ) is 1 + δ( + 1). This is a consequence of the fact that Therefore, there is in tree δ ( ) a leaf of index +1 for any value ∈ δ( + 1)]. Hence, and due to the fact that by construction, tree δ ( ) is a δ-increasing tree, the map tree δ is well-defined. Now, let φ be the map from the set of all δ-increasing trees of size to Cl δ ( ) defined recursively as follows. If t is the leaf, set φ(t) := . Otherwise, consider the node with the maximal label in t. Since t is increasing, this node has no children. Set t ′ as the δ-increasing tree obtained by replacing this node by a leaf in t, and set as the index of the leaf of t ′ on which this maximal node of t is attached (this index is 1 if t ′ is the leaf). Then, set φ(t) := φ(t ′ )( − 1). The statement of the proposition follows by showing by induction on that φ is the inverse of the map tree δ .
In [CP19], -decreasing trees are considered, where is a sequence of length 0 of nonnegative integers. These trees are labeled decreasingly and any internal node labeled by ∈ [ ] has arity . As a consequence of Proposition 1.1.1, any -decreasing tree can be encoded by a δ-increasing tree where δ is a rooted weakly increasing range map satisfying The correspondence between such -decreasing trees and δ-increasing trees consists in relabeling by + 1 − each internal node labeled by ∈ [ ]. A consequence of all this is that δ-cliffs can be seen as generalizations of -decreasing trees by relaxing the considered conditions on δ.
1.2. δ-cliff posets. We endow now the set of all δ-cliffs of a given size with an order relation, give some recalls about poset and lattice theory, and establish a link between the poset of 1-cliffs and the weak Bruhat order of permutations.
1.2.1. First definitions. Let δ be a range map and be the partial order relation on Cl δ defined by for any ∈ Cl δ such that | | = | | and for all ∈ [| |]. For any 0, the poset (Cl δ ( ) ) is the δ-cliff poset of order . Figure 1 shows the Hasse diagrams of some δ-cliff posets.

F
. Hasse diagrams of some δ-cliff posets. Let us introduce some notation about δ-cliffs. For any ∈ Cl δ ( ) and ∈ [ ], let ↓ ( ) (resp. ↑ ( )) be the word on Z of length obtained by decrementing (resp. incrementing) by 1 the -th letter of . Let also, for any ∈ Cl δ ( ), D( ) := { ∈ [ ] : = } be the set of all indices of different letters between and . For any ∈ Cl δ ( ), let ∧ be the δ-cliff of size defined for any ∈ [ ] by ( ∧ ) := min{ } We also define ∨ similarly by replacing the min operation by max in the previous definition. For any ∈ Cl δ ( ), the difference between and is the word − on Z of length defined for any ∈ [ ] by ( − ) := − Observe that when , − is a δ-cliff. The δ-complementary c δ ( ) of ∈ Cl δ ( ) is the δ-cliff1 δ ( ) − For instance, by setting := 0010, if is seen as a 1-cliff, then c 1 ( ) = 0113, and if is seen as a 2-cliff, then c 2 ( ) = 0236. This map c δ is an involution.

First properties and recalls.
A study of the 1-cliff posets appears in [Den13]. Our definition stated here depending on δ is therefore a generalization of these posets. The structure of the δ-cliff posets is very simple since each of these posets of order is isomorphic to the Cartesian product δ(1)] × · · · × δ( )], where ] is the total order on + 1 elements. It follows from this observation that each δ-cliff poset is a lattice admitting respectively ∧ and ∨ as meet and join operations. Recall that a lattice ( Recall also that all distributive lattices are modular and graded [Sta11]. Since total orders are distributive and the distributivity is preserved by the Cartesian product of posets, Cl δ ( ) is a distributive lattice.
Recall that the covering relation of a poset is the set of all pairs ( ) ∈ 2 such that the interval [ ] has cardinality 2. It follows immediately from the definition of that the covering relation ⋖ of Cl δ ( ) satisfies ⋖ if and only if there is an index ∈ [ ] such that = ↑ ( ). Moreover, these posets Cl δ ( ) are graded, and the rank of a δ-cliff is ω( ). The least element of the poset is0 δ ( ) while the greatest element1 δ ( ).
Let us make some reminders about poset morphisms used thereafter. If ( 1 1 ) and ( 2 2 ) are two posets, a map φ : 1 → 2 is a poset morphism if for any ∈ 1 , 1 implies φ( ) 2 φ( ). We say that 2 is an order extension of a poset 1 if there is a map φ : 1 → 2 which is both a bijection and a poset morphism. A map φ : 1 → 2 is a poset embedding if for any ∈ 1 , 1 if and only if φ( ) 2 φ( ). Observe that a poset embedding is necessarily injective. A map φ : 1 → 2 is a poset isomorphism if φ is both a bijection and a poset embedding.
1.2.3. Links with the weak Bruhat order. Let S be the graded set of all permutations where the degree of a permutation is its length as a word. A coinversion of a permutation σ is a pair σ σ such that σ < σ and < . For any 0, the weak Bruhat order of order is a partial order (S( ) S ) wherein for any σ ν ∈ S( ), σ S ν if the set of all coinversions of σ is contained in the set of all coinversions of ν. By denoting by , ∈ [ −1], the -th elementary transposition, the covering relation ⋖ S of this poset satisfies σ ⋖ S σ for any σ ∈ S( ) and any ∈ [ − 1] such that σ < σ +1 .
When δ is a rooted weakly increasing range map, let us consider the binary relation ⋖ ′ on Cl δ ( ) defined as follows. Let ∈ Cl δ and t := tree δ ( ). We have ⋖ ′ if there is an index ∈ [ ] such that = ↑ ( ) and all the children of the node labeled by of t are leaves, except possibly the first of its brotherhood. For instance, for δ := 0233579 ω and the δ-cliff := 021042, since we observe that all the children of the nodes labeled by 2, 3, 5 and 6 are leaves, except possibly the first ones. For this reason, is covered by ↑ 3 ( ) = 022042 and by ↑ 6 ( ) = 021043, but not by ↑ 2 ( ) = 031042 since this word is not a δ-cliff.
The reflexive and transitive closure ′ of this relation is an order relation. By Proposition 1.1.1, this endows the set of all δ-increasing trees with a poset structure. It follows immediately from the description of the covering relation ⋖ of Cl δ ( ) provided in Section 1.2.2 that ⋖ ′ is a refinement of ⋖. For this reason (Cl δ ( ) ) is an order extension of (Cl δ ( ) ′ ). Figure 2 shows an example of a Hasse diagram of such a poset. Proof. Let φ be the map from the set of all words of size of integers without repeated letters to the set of increasing binary trees of size where internal nodes are bijectively labeled by the letters of , defined recursively as follows. If σ is the empty word, then φ(σ ) is the leaf. Otherwise, σ decomposes as σ = ′ where is the least letter of σ , and and ′ are words of integers. In this case, φ(σ ) is the binary tree consisting in a root labeled by and having as left subtree φ( ′ ) and as right subtree φ( ) -observe the reversal of the order between and ′ . Now, by induction on , one can prove that for any permutation σ of size , the binary trees φ(σ ) and tree 1 (leh(σ )) are the same.
Assume that σ and ν are two permutations such that σ ⋖ S ν. Thus, by definition of ⋖ S , σ decomposes as σ = ′ and ν as ν = ′ where and are letters such that < , and and ′ are words of integers. By definition of φ, since and are adjacent in σ , the right subtree of the node labeled by of φ(σ ) is empty. Therefore, due to the property stated in the first part of the proof, and by definition of the map tree 1 and of the covering relation ⋖ ′ , one has leh(σ ) ⋖ ′ leh(ν). Conversely, assume that and are two 1-cliffs such that ⋖ ′ . Thus, by definition of ⋖ ′ , is obtained by changing a letter , 2, in by + 1, and in tree 1 ( ), the right subtree of the node labeled by is empty. Let σ := leh −1 ( ) and ν := leh −1 ( ). Since φ(σ ) and tree 1 ( ) are the same increasing binary trees, we have, from the definition of the map φ, that −1 < . Finally, by definition of ⋖ S , one obtains σ ⋖ S ν.
We have shown that the bijection leh between S( ) and Cl 1 ( ) is such that, for any σ ν ∈ S( ), σ ⋖ S ν if and only if leh(σ ) ⋖ ′ leh(ν) . For this reason, leh is a poset isomorphism. Therefore, Proposition 1.2.1 says in particular that the 1-cliff poset is an extension of the weak Bruhat order. Besides, for all rooted weakly increasing range maps δ, one can see (Cl δ ( ) ′ ) as generalizations of the weak Bruhat order. Supported by some computer experiments, we state the following conjecture.
Observe that when is spread, each poset ( ), 0, is bounded, that is it admits a least and a greatest element. Observe also that if is both minimally and maximally extendable, then is spread. . Since is coated, belongs to , and moreover, since is maximal, . Therefore, #D( ) = 1. This proves that there exists a ′ ∈ ( ) such that ⋖ ′ and #D( ′ ) = 1. Thus, is straight.
We use here the notion of -th dimension of range maps, defined in Section 1.1.1. In the case where is straight, we define the graded set of ⋆ input-wings as the set ( ) containing any ∈ which covers exactly dim | | (δ) elements; ⋆ output-wings as the set ( ) containing any ∈ which is covered by exactly dim | | (δ) elements; ⋆ butterflies as the set ( ) being the intersection ( ) ∩ ( ).
Equivalently, ∈ is an input-wing (resp. output-wing) if it is possible to decrement (resp. increment) all values at all positions ∈ [| |] such that δ( ) = 0. Observe also that if there is an 1 such that δ( ) = 1, there are no butterfly in ( ) for all .
1.3.1. EL-shellability. Let ( ) and (Λ Λ ) be two posets, and λ : ⋖ → Λ be a map (here ⋖ is seen as the set of all pairs ( ) such that covers in ). For any saturated chain (1) ( ) of , by a slight abuse of notation, we set We say that a saturated chain of is λ-increasing (resp. λ-weakly decreasing) if its image by λ is an increasing (resp. weakly decreasing) word w.r.t. the partial order relation Λ . We say also that a saturated chain (1) ( ) of is λ-smaller than a saturated chain (1) of if the image by λ of (1) ( ) is smaller than the image by λ of (1) (ℓ) for the lexicographic order induced by Λ . The map λ is an EL-labeling of if there exist such a poset Λ and a map λ such that for any ∈ satisfying , there is exactly one λincreasing saturated chain from to which is minimal among all saturated chains from to w.r.t. the order on saturated chains just described. The poset is EL-shellable [BW96,BW97] if is bounded and admits an EL-labeling.
The EL-shellability of a poset implies several topological and order theoretical properties of the associated order complex ∆( ) made of all the chains of . For instance, one of the consequences for for having at most one λ-weakly decreasing chain between any pair of its elements is that the Möbius function of takes values in {−1 0 1}. In an equivalent way, the simplicial complex associated with each open interval of is either contractile or has the homotopy type of a sphere [BW97].
For the sequel, we set Λ as the poset Z 2 wherein elements are ordered lexicographically. For any straight graded subset of Cl δ , let us introduce the map λ : ⋖ → Z 2 defined for any Observe that the fact that is straight ensures that λ is well-defined.
Theorem 1.3.2. Let δ be a range map and be a coated graded subset of Cl δ . For any 0, the map λ is an EL-labeling of ( ). Moreover, there is at most one λ -weakly decreasing chain between any pair of elements of ( ).
Proof. By Lemma 1.3.1, the fact that is coated implies that is also straight. Let ∈ ( ) such that . Since is straight, the image by λ of any saturated chain from to is well-defined. Now, let = (0) (1) ( ) = be the sequence of elements of ( ) defined in the following way. For any ∈ − 1], the word ( +1) is obtained from ( ) by increasing by the minimal possible value 1 the letter ( ) such that is the greatest index satisfying ( ) < .
By construction, for any ∈ − 1], each ( +1) writes as ( +1 , where is some positive integer. There is at least one value such that ( ) belongs to ( ) since by hypothesis, is coated. For this reason, the considered sequence is a well-defined saturated chain in ( ). This saturated chain is also λ -increasing by construction. Moreover, since is straight, if one consider another saturated chain from to , this chain passes through a word obtained by incrementing a letter which has not a greatest index, and one has to choose later in the chain the letter of the smallest index to increment it. For this reason, this saturated chain would not be λ -increasing.
Assume now that there exists in a λ -weakly decreasing saturated chain of the form = (0) (1) ( ) = By definition of λ and of the poset Λ, for any ∈ − 1], the word ( +1) is obtained from ( ) by increasing by the minimal possible value the letter ( ) such that is the smallest index satisfying ( ) < . If it exists, this saturated chain is by construction the unique λ -weakly decreasing saturated chain from to .
1.3.2. Meet and join operations, sublattices, and lattices. Here we give some sufficient conditions on for the fact that each ( ), 0, is a lattice.
Second, assume instead that is minimally extendable. Observe that the fact that is minimally extendable ensures that ⇓ is a well-defined map. Let also, for any 0 and ∈ ( ), ∧ When is maximally extendable, we denote by ⇑ the -incrementation map defined in the same way as the -decrementation map with the difference that in the previous definitions, the operation max is replaced by the operation min and the relation is replaced by the relation . Here, the fact that is maximally extendable ensure that ⇑ is well-defined. We also define the operation ∨ in the same way as ∧ with the difference that in the previous definitions, the map ⇓ is replaced by ⇑ and the operation ∧ is replaced by the operation ∨. Theorem 1.3.3. Let δ be a range map and be a closed by prefix and minimally (resp. maximally) extendable graded subset of Cl δ . The operation ∧ (resp. ∨ ) is, for any 0, the meet (resp. join) operation of the poset ( ).
Proof. Let us show the property of the statement of the theorem in the case where is minimally extendable. The other case is symmetric. We proceed by induction on 0.
When = 0, the property is trivially satisfied. Let 1 and ∈ ( ). Since is closed by prefix, one has = ′ and = ′ with ′ ′ ∈ ( − 1) and ∈ N. Since is minimally extendable, where := max{ min{ } : ⇓ ( ′ ∧ ′ ) ∈ } Now, by induction hypothesis, we obtain where ∧ is the meet operation of the poset ( − 1). First, we deduce from the above computation that for any ∈ [ ], the -th letter of ∧ is nongreater than min{ }, and that ∧ belongs to ( ). Therefore, ∧ is a lower bound of { }. Second, by induction hypothesis, ′ := ′ ∧ ′ is the greatest lower bound of { ′ ′ }. By construction, since is the greatest letter such that , , and ′ ∈ holds, any other lower bound of { } is smaller than ′ . This prove that ′ is the greatest lower bound of { } and implies the statement of the theorem.

Join-irreducible elements.
Recall that an element of a lattice is join-irreducible (resp. meet-irreducible) if covers (resp. is covered by) exactly one element in . We denote by J( ) (resp. M( )) the set of join-irreducible (resp. meet-irreducible) elements of . These notions are usually considered specially for lattices but we can take the same definitions even when is just a poset. Proof. Assume first that is a join-irreducible element of ( ). Then, there is exactly one element ′ of such that ′ ⋖ . Since is straight, #D( ′ ) = 1, implying that satisfies the stated condition.
This operation has been introduced in [Day92] as an operation on posets preserving the property to being a lattice. On the other way round, we say that is obtained by an interval contraction from a poset ′ if there is an interval I of such that [I] is isomorphic as a poset to ′ [CLCdPBM04].
A lattice is constructible by interval doubling (spelled as "bounded" in the original article) if is isomorphic as a poset to a poset obtained by performing a sequence of interval doubling from the singleton lattice. It is known from [Day79] that such lattices are semi-distributive. Recall that a finite lattice is constructible by interval doubling if and only if it is congruence uniform, and then in particular, the number of join-irreducible elements of determines the number of interval doubling steps needed to create (see [Day79] and [Müh19]).
The aim of this section is to introduce a sufficient condition on a graded subset of Cl δ for the fact that each ( ), 0, is constructible by interval doubling. We shall moreover describe explicitly the sequence of interval doubling operations involved in the construction of ( ) from the trivial lattice.
Let be a nonempty subposet of Cl δ ( ) for a given fixed size 1. Let us denote by m( ) the letter max{ : ∈ }. For any ∈ δ( )], let := { ∈ : = } and Observe that is a subposet of while may contain δ-cliffs that do not belong to . The derivation of is the set In other words, ( ) is the set of all the cliffs obtained from by decrementing their last letters if they are equal to m( ) or by keeping them as they are otherwise. Observe that ( ) is not necessarily a subposet of . Nevertheless, ( ) is still a subposet of Cl δ ( ). Observe also that m( ( )) m( ) − 1 For instance, by considering the subposet This definition still holds when m( ) = 0. Observe that any δ-cliff 0 −1 , 1, of covers exactly the single element 0 −1 ( − 1) of . This element exists by (N1). Therefore, when is a lattice, these δ-cliffs are join-irreducible. Lemma 1.3.5. Let δ be a range map and be a nonempty subposet of Cl δ ( ) for an 1.
If is nested, then for any ∈ m( )], is an interval of .
Proof. First, by (N1), admits 0 −1 as unique least element. It remains to prove that has at most one greatest element. By contradiction, assume that there are in two different greatest elements and , where ∈ Cl δ ( − 1). Then, by setting := m( ), in the δ-cliffs and are still incomparable. Since these two elements are also greatest elements of , this implies that is not an interval in . This contradicts (N2).
Let δ be a range map and be a nonempty subposet of Cl δ ( ) for an 1.
If m( ) 1 and is nested, then ( ) is nested.
is an interval of . Due to the fact ′ − 1, one has = ′ , so that ′ is an interval of . This is equivalent to the fact that ′ ′ is an interval of ′ . By Lemma 1.3.6, the relation ′ ′ = ′ holds and leads to the fact that ′ ′ is an interval of ′ ′ . Therefore, ′ satisfies (N2). Lemma 1.3.8. Let δ be a range map and be a nonempty subposet of Cl δ ( ) for an 1.

If m( ) 1 and is nested, then is isomorphic as a poset to ( )[I]
where I is the interval m( )−1 of ( ).
Since I is an interval of ′ , we can now consider the poset ′ [I]. By definition of the interval doubling operation, This map φ is well-defined because, respectively, one has ′ = for any ∈ ′ − 1], Lemma 1.3.6 holds, I is in particular a subset of , and satisfies (N2). Let now ψ : → ′ [I] be the map satisfying By similar arguments as before, this map ψ is well-defined. Moreover, by construction, ψ is the inverse of φ. Therefore, φ is a bijection. The fact that φ is a poset embedding comes by definition of φ and from the fact that, due to the property of to be nested, for any ′ ∈ ′ \ ′ , all elements greater than ′ in ′ do not belong to ′ . Thus, ′ [I] is isomorphic as a poset to .
By assuming that is nested, the sequence of derivations from is the sequence Given a graded subset of Cl δ , we say by extension that is nested if for all 0, the posets ( ) are nested.
Theorem 1.3.9. Let δ be a rooted range map and be a nested and closed by prefix graded subset of Cl δ . For any 1, ( ) is constructible by interval doubling. Moreover, is a sequence of interval contractions from ( ) to the trivial lattice { }.
Proof. We proceed by induction on 0. If = 0, since δ is rooted, we necessarily have (0) ≃ { }, and this poset is by constructible by interval doubling. Assume now that 1 and set := ( ). Since is nested, the sequence of reductions from is well-defined. By Lemmas 1.3.7 and 1.3.8, by setting ′ := m( ) ( ) is obtained by performing a sequence of interval doubling from the poset ′ . Now, due to the definition of the derivation algorithm , ′ is made of the δ-cliffs of wherein the last letters have been replaced by 0. This poset ′ is therefore isomorphic to the poset ′′ formed by the prefixes of length − 1 of . Since is closed by prefix, ′′ is thus the poset ( − 1). By induction hypothesis, this last poset is constructible by interval doubling. Therefore, ( ) also is. All this produces the sequence (1.3.10) of interval contractions. 1.3.5. Elevation maps. We introduce here a combinatorial tool intervening in the study of the three Fuss-Catalan posets introduced in the sequel.
Let be a closed by prefix graded subset of Cl δ . For any ∈ , let By definition, F ( ) is the set of all the letters that can follow to form an element of . For any 0, the -elevation map is the map e : ( ) → Cl δ ( ) defined, for any ∈ ( ) From an intuitive point of view, the value of the -th letter of e ( ) is the number of cliffs of obtained by considering the prefix of ending at the letter and by replacing this letter by a smaller one. Remark in particular that e Cl δ is the identity map. Besides, we say that any ∈ is an exuviae if e ( ) = .
Let be the graded set wherein for any 0, ( ) is the image of ( ) by the -elevation map. We call this set the -elevation image. Observe that is a graded subset of Cl δ . Note also that for any ∈ , e ( ) .

Proposition 1.3.10. Let δ be a range map and be a closed by prefix graded subset of Cl δ . For any 0, the -elevation map is injective on the domain ( ).
Proof. We proceed by induction on . When = 0, the property is trivially satisfied. Let ∈ ( ) such that 1 and e ( ) = e ( ) Since is closed by prefix, we have = ′ and = ′ where ′ ′ ∈ ( −1) and ∈ N. By definition of e , we have e ( ′ ) = e ( ′ ) and e ( ′ ) = e ( ′ ) where ∈ N. Hence, e ( ′ ) = e ( ′ ) which leads, by induction hypothesis, to the fact that ′ = ′ . Moreover, we deduce from this and from the definition of the -elevation map that there are exactly letters ′ smaller than such that ′ ′ ∈ and that there are exactly letters ′ smaller than such that ′ ′ ∈ . Therefore, we have = and thus = , establishing the injectivity of e . Lemma 1.3.11. Let δ be a range map and be a closed by prefix graded subset of Cl δ . The -elevation image is closed by prefix.
1.3.6. Geometric cubic realizations. Let be a graded subset of Cl δ . For any 0, the realization of ( ) is the geometric object C( ( )) defined in the space R and obtained by placing for each ∈ ( ) a vertex of coordinates ( 1 ), and by forming for each ∈ ( ) such that ⋖ an edge between and . Remark that the posets of Figure 1 represent actually the realizations of δ-cliff posets. We will follow this drawing convention for all the next figures of posets in all the sequel. When is straight, every edge of C( ( )) is parallel to a line passing by the origin and a point of the form (0 0 1 0 0). In this case, we say that C( ( )) is cubic.
As a side remark, we would like to stress that the present notion of geometric realization of a poset differs from the usual one saying that it consists in the geometric realization of the simplicial complex of the chain of the poset.
Let us assume from now that is straight. Let ∈ ( ) such that . The word is cell-compatible with if for any word of length such that for any ∈ [ ], ∈ { }, then ∈ . In this case, we call cell the set of points By definition, a cell is an orthotope, that is a parallelotope whose edges are all mutually orthogonal or parallel. For any 0, the -volume vol (C( ( ))) of C( ( )) is the volume obtained by summing the volumes of all its all its cells of dimension , computed by not counting several times potential intersecting orthotopes. The volume vol(C( ( ))) of C( ( )) is defined as vol (C( ( ))) where is the largest integer such that C( ( )) has at least one cell of dimension . Figure 3 shows examples of these notions. Figure 3a shows a cubic realization wherein 00 is cell-compatible with 12. Hence, 00 12 is a cell. The point 1 2 3 2 ∈ R 2 is inside 00 12 , and since there are no elements of the poset inside the cell, this cell is pure. Figure 3b shows a cubic realization wherein 00 is not cell-compatible with 22 because 02 does not belong to the poset. Nevertheless, 00 11 , 10 21 , and 11 22 are pure cells of dimension 2. Figure 3c shows a cubic realization wherein 00 22 is a non-pure cell. Indeed, the δ-cliff 11 is an element of the poset and is inside this cell. Finally, Figure 3d shows a cubic realization having 1 as  There is a close connection between output-wings (resp. input-wings) of ( ), 0, and the computation of the volume of C( ( )): if is a cell of maximal dimension of C( ( )), then due to the fact that is straight, (resp. ) is an output-wing (resp. input-wing) of ( ). When for any 0, then the volume of C( ( )), 0, writes as When some cells of { ρ( ) : ∈ ( )( )} intersect each other, the expression for the volume would not be at as simple as (1.3.15) and can be written instead as an inclusion-exclusion formula. Of course, the same property holds when ρ is instead a map from ( )( ) to ( )( ) by changing accordingly the previous text.
Recall that the order dimension [Tro92] of a poset is the smallest nonnegative integer such that there exists a poset embedding of into N where is the componentwise partial order. Recall that the hypercube of dimension 0 is the poset on the set of the subsets of [ ] ordered by set inclusion. It can be shown that the order dimension of is .
Proposition 1.3.13. Let δ be a range map and be a straight graded subset of Cl δ . If, for an 0, C( ( )) has a cell of dimension dim (δ), then the order dimension of the poset . This poset has order dimension dim (δ), so that the order dimension of ( ) is at most dim (δ). Besides, since is straight, the notion of cell is welldefined in the cubic realization of ( ). By hypothesis, ( ) contains a cell of dimension dim (δ). Thus, there is a poset embedding of dim (δ) into the interval [ ] of ( ). Therefore, the order dimension of ( ) is at least dim (δ).
As a particular case of Proposition 1.3.13, the order dimension of Cl δ ( ) is dim (δ). This explains the terminology of " -th dimension of δ" for the notation dim (δ) introduced in Section 1.1.1.

S F -C
We present here some examples of subposets of δ-cliff posets. We focus in this work on three posets whose elements are enumerated by -Fuss-Catalan numbers for the case δ = m, 0. We provide some combinatorial properties of these posets like among others, a description of their input-wings, output-wings, and butterflies, a study of their order theoretic properties, and a study of their cubic realizations. We end this section by establishing links between these three families of posets in terms of poset morphisms, poset embeddings, and poset isomorphisms. We shall omit some straightforward proofs (for instance, in the case of the descriptions of input-wings, output-wings, butterflies, meet-irreducible and join-irreducible elements of the posets -the descriptions of these two families use Proposition 1.3.4).
We use the following notation conventions. Poset morphisms are denoted through arrows , poset embeddings through arrows , and poset isomorphisms through arrows .
2.1. δ-avalanche posets. We begin by introducing a first Fuss-Catalan family of posets. As we shall see, these posets are not lattices but they form an important tool to study the next two families of Fuss-Catalan posets. ∈ Av δ ( ). Since δ( + 1) δ( ), 0 is a δ-avalanche. This establishes (ii). Finally, we have immediately that Av 0 ω is maximally extendable. Moreover, when δ = 0 ω , there is an 1  Let δ be a weakly increasing range map. Notice that in general, Av δ ( ) is not bounded. Since for all ∈ Av δ ( ), ω( ) δ( ), we have ∈ max Av δ ( ) if and only if ω( ) = δ( ). Moreover, due to the fact that any δ-cliff obtained by decreasing a letter in a δ-avalanche is also a δ-avalanche, the poset Av δ ( ) is the order ideal of Cl δ ( ) generated by max Av δ ( ). Finally, as a particular case, we shall show as a consequence of upcoming Proposition 2.2.7 that for any 0 and 1, # max Av m ( ) = cat ( − 1).
Proposition 2.1.3. For any weakly increasing range map δ and 0, the poset Av δ ( ) (iii) is graded, where the rank of an avalanche is its weight; (iv) admits an EL-labeling; and are two δ-avalanches of size such that , then for any is a δ-avalanche. For this reason, (ii) checks out. Point (iv) follows from (ii), and Theorem 1.3.2. The four posets of avalanches, input-wings, output-wings, and butterflies are linked in the following way.  Now, by using the fact that ( ) satisfies ( ) = 1 + ( ) +1 we have This relation satisfied by ′ ( ) between the first and last members of (2.1.5) is known to be the one of the generating series of twisted -Fuss-Catalan numbers (see [Slo] for instance).
By Proposition 2.1.6, the first numbers of output-wings of Av m ( ) by sizes are The fourth sequence is Sequence A006013 of [Slo]. As a side remark, for any 1, the generating series of the graded set (Av m ) is 1 plus the inverse, for the functional composition of series, of the polynomial (1 − ) . Proof. Point (i) is an immediate consequence of the definition of δ-hills. We have immediately that Hi 0 ω is minimally extendable. Moreover, when δ = 0 ω , there is an 1 such that δ( ) 1.
Since is in particular a δ-cliff of size , then δ( ), so that ∈ Av δ ( ). This shows that      < m ′ ( ). A possible bijection between these two sets sends any ∈ (Hi m )( ) to the m ′ -cliff of the same size such that for any ∈ [ ], = − + 1. We have already seen in the proof of Proposition 2.1.6 that these sets are in one-to-one correspondence with ( − 1)-Dyck paths which cannot be written as a nontrivial concatenation of two ( − 1)-Dyck paths. Therefore, the statement of the proposition follows. where in an input-wing, is by definition of 2 equal to 1. Therefore, vol(C(Hi m ( ))) is equal to the number of input-wings of Hi m ( ). The statement of the proposition follows now from Theorem 2.2.5.

δ-canyon posets.
We introduce here our last family of posets. They are defined on particular δ-cliffs called δ-canyons. As we shall see, under some conditions these posets are lattices but not sublattices of δ-cliff lattices. ⋆ If α < 0, then we say that and are independent in (graphically, the diagonal of falls under the -axis before reaching the segment of ); ⋆ If α ∈ − 1], then we say that is hindered by in (graphically, the diagonal of hits the segment of ); ⋆ If α , then we say that dominates in (graphically, the segment of is below or on the diagonal of ).

Proposition 2.3.1. For any range map δ, the graded set Ca δ is (i) closed by prefix; (ii) is minimally extendable; (iii) is maximally extendable if δ is increasing.
Proof. Let be a δ-canyon of size 0. Immediately from the definition of the δ-canyons, it follows that 0 is a δ-canyon of size + 1, and that for any prefix ′ of , ′ is a δ-canyon. Therefore, Points (i) and (ii) check out. Let us now consider the δ-cliff ′ := δ( + 1). If δ is increasing, for all ∈ [ ], +1− +1 − . Therefore, ′ is a δ-canyon. Therefore, (iii) holds.
Let us now introduce a series of definitions and lemmas in order to show that the sets Ca δ ( ) and Hi δ ( ) are in one-to-one correspondence when δ is an increasing range map. Proof. Assume that is of size and set := d( ). Assume that is a δ-canyon for a letter ∈ N. Then, the index + 1 is hindered by no other index in . Since is obtained by changing to 0 some letters of , the index + 1 remains hindered by no other index in . Therefore, is also a δ-canyon. Conversely, assume that is a δ-canyon for a letter ∈ N. Then, the index + 1 is hindered by no other index in . By contradiction, assume that is not a δ-canyon. This implies that the index + 1 is hindered by an index in . Let us take maximal among all indices satisfying this property. Due to the maximality of , is dominated by no other index in so that we have = . This implies that + 1 is hindered by in , which contradicts our hypothesis. Therefore, is a δ-canyon.
Proposition 2.3.6. For any increasing range map δ and any 0, the map e −1 Hi δ • e Ca δ is a bijection between Ca δ ( ) and Hi δ ( ).
Proof. First, since δ is increasing, by Propositions 2.2.1 and 2.3.1, both Hi δ and Ca δ are closed by prefix. Therefore, the maps e Hi δ and e Ca δ are well-defined. By Proposition 1.3.10, the maps e Ca m and e Hi m are injective, and by Propositions 2.2.2 and 2.3.5, they both share the same image Av m ( ). This implies that e Ca m is a bijection from Ca m to Av m ( ), and that e −1 Hi m is a well-defined map and is a bijection from Av m ( ) to Hi m ( ). Therefore, the statement of the proposition follows.  Tamari diagrams and have been introduced in [Pal86]. The set of these objects of size is in one-to-one correspondence with the set of binary trees with internal nodes. It is also known that the componentwise comparison of Tamari diagrams is the Tamari order [Pal86]. Recall that the Tamari order admits, as covering relation, the right rotation operation in binary trees. It has also the nice property to endow the set of binary trees of a given size with a lattice structure [HT72]. Besides, a study of the posets of the intervals of the Tamari order, based upon a generalization of Tamari diagrams, has been performed in [Com19]. The Tamari posets admit a lot of generalizations, for instance through the so-called -Tamari posets [BPR12] where 0, and through the ν-Tamari posets [PRV17] where ν is a binary word. Our δ-canyon posets can be seen as different generalizations of Tamari posets. For any 2, the m-canyon posets are not isomorphic to the -Tamari posets. Moreover, we shall prove in the sequel that for any increasing map δ, Ca δ is a lattice. As already mentioned, Tamari posets have the nice property to be lattices [HT72], are also EL-shellable [BW97], and constructible by interval doubling [Gey94]. The same properties hold for -Tamari lattices, see respectively [BMFPR11] and [Müh15] for the first two ones. The last one is a consequence of the fact that -Tamari lattices are intervals of 1-Tamari lattices [BMFPR11] and the fact that the property to be constructible by interval doubling is preserved for all sublattices of a lattice [Day79]. As we shall see here, the δ-canyon posets have the same three properties.  These computations of the join of two elements are similar to the ones described in [Mar92] (see also [Gey94]) for Tamari lattices.
Besides, as pointed out by Proposition 2.3.7, when δ is an increasing range map, each Ca δ ( ) is constructible by interval doubling. Figure 9 shows a sequence of interval contractions performed from Ca 2 (4) in order to obtain Ca 2 (3).  Let be an m-cliff satisfying ∈ { ρ( ) } for any ∈ [ ]. Since ρ ′ is the inverse map of ρ, this is equivalent to the fact that where is the input-wing ρ( ) of Ca m ( ). Therefore, by definition of ρ ′ , 1 = 0 and ∈ − 1] for any ∈ [2 ]. The fact that is an input-wing implies, by Proposition 2.3.8, that < +1 for all ∈ [ − 1]. This implies that is an m-canyon, so that (ii) checks out.
Point (iii) follows directly from the definition of ρ: since ρ( ) is obtained by incrementing all the letters of , except the first, in a minimal way so that the obtained m-cliff is an m-canyon, there cannot be any m-canyon inside the cell ρ( ) .
Finally, assume that there are two input-wings and of Ca m ( ) such that there is a point := ( 1 ) ∈ R such that is inside both the cells ρ ′ ( ) and ρ ′ ( ) . By contradiction, let us assume that = and let us set ∈ [2 ] as the smallest position such that = . Therefore, we have in particular This contradicts our hypothesis and shows that = . Therefore, (iv) holds. This algorithm ρ brought by Proposition 2.3.10 describes the cells of maximal dimension of the cubic realization of Ca m ( ). The definition of ρ is inspired by an analogous algorithm introduced by the first author in [Com19] to describe the cells of a geometric realization of the lattices of Tamari intervals. Figure 10 shows some examples of images of output-wings of Ca m ( ) by ρ.
The three posets of the input-wings, output-wings, and butterflies of canyon posets are linked in the following way.

Poset morphisms and other interactions.
The purpose of this part is to state the main links between the three posets Av δ , Hi δ , and Ca δ when δ is an increasing range map. We shall also consider their subposets formed by their input-wings, output-wings, and butterflies elements in the particular case where δ = m for an 0.

Order extensions.
Observe that the map e Ca δ is not a poset morphism. Indeed, for instance in Ca 1 one has 002 012 but e Ca 1 (002) = 002 011 = e Ca 1 (012) Nevertheless, by composing this map on the left with the inverse of the Hi δ -elevation map, we obtain a poset morphism, as stated by the next theorem.
Lemma 2.4.1. Let δ be a range map, and and be two δ-canyons of size . If , then ω(e Ca δ ( )) ω(e Ca δ ( )) Proof. First, since by Proposition 2.3.1, Ca δ is closed by prefix, e Ca δ is well-defined. By considering the contrapositive of the statement of the lemma and by Lemma 2.3.4, we have to show that for any δ-canyons and of size , ω(d( )) > ω(d( )) implies that there exists ∈ [ ] such that > . We proceed by induction on . If = 0, the property holds immediately. Assume now that = ′ and = ′ are two δ-canyons of size + 1 such that ω(d( ′ )) > ω(d( ′ )) where ′ and ′ are δ-canyons of size and ∈ N. If ω(d( ′ )) > ω(d( ′ )), then by induction hypothesis, there is ∈ [ ] such that ′ > ′ . Since = ′ and = ′ , the property holds.
Even if, by Proposition 2.3.6, e −1 Hi δ •e Ca δ : Ca δ ( ) → Hi δ ( ) is a bijection, this map is not a poset isomorphism. This is the case since there does not exist for instance a poset isomorphism between Ca 1 (3) and Hi 1 (3) -their Hasse diagrams are not superimposable. Moreover, as a consequence of Theorem 2.4.2, for any 0, Hi δ ( ) is an order extension of Ca δ ( ).
Furthermore, it is possible to show by induction on the length of the δ-canyons and by using Lemma 2.3.3 that Ca δ satisfies the prerequisites of Proposition 1.3.12. Therefore, Ca δ ( ) is an order extension of Av δ ( ).
To summarize the situation, when δ is an increasing range map, the three families of Fuss-Catalan posets fit into the chain of posets for the order extension relation. This phenomenon is analogous to the one stating that Stanley lattices are order extensions of Tamari lattices, which in turn are order extension of Kreweras lattices [Kre72] (see for instance [BB09]). In the present case, avalanche posets play the role of the Kreweras lattices, canyon posets play the role of the Tamari posets, and hill posets play the role of Stanley posets. The difference is that for δ = 1, even if canyon lattices coincide with Tamari lattices and hill lattices coincide with Stanley lattices, the avalanche posets are not isomorphic to Kreweras lattices. Figure 12 gives an example of an instance of (2.4.1).

Isomorphisms between subposets.
There is a link between the hill and the canyon posets, as stated by the following result.  . The subposet of Ca 2 (4) formed by its input-wings is isomorphic to Hi 1 (4).
any 1 and 0 the diagram of poset morphisms, embeddings, or isomorphisms.

A δ-
This part of the work is devoted to endow the sets of δ-cliffs with algebraic structures. We describe a graded associative algebra on δ-cliffs motivated by a connection with the δ-cliff posets. Indeed, the product of two δ-cliffs is a sum of δ-cliffs forming an interval of a δ-cliff poset. This property is shared by a lot of combinatorial and algebraic structures. For instance, the algebra FQSym of permutations is related to the weak Bruhat order [DHT02,AS05], the algebra PBT of binary trees is related to the Tamari order [LR02,HNT05], and the algebra Sym of integer compositions is related to the hypercube [GKL + 95].
3.1. Coalgebras and algebras. We introduce here a cograded coalgebra structure on the linear span of all δ-cliffs and then, by considering the dual structure, we obtain a graded algebra. When δ satisfies some properties, this gives an associative algebra.
From now, K is the ground field (of characteristic zero) of all algebraic structures to be defined next. For any graded vector space , we denote by ( ) the Hilbert series of .
We use here the notions of valleys in range maps and of valley-free range maps, defined in Section 1.1.1. Let ∆ : Cl δ → Cl δ ⊗ Cl δ be the cobinary coproduct defined, for any ∈ Cl δ , by where N * denotes the set of all words on N. This coproduct is well-defined since any prefix of a δ-cliff is a δ-cliff and the image of a word on N by the δ-reduction map is by definition a δ-cliff. For instance, for δ := 1221013 ω , we have in Cl δ , Proof. The first part of the statement is a direct consequence of the definition of ∆.
To establish the second part, let us compute the two ways to apply twice the coproduct ∆ on a basis element of Cl δ . For any ∈ Cl δ , we have .1.4) and where for any 0, δ is the range map satisfying δ ( ) = min{δ( ) δ( + )} for any 1.
Proof. Assume first that := ∈ Cl δ . By Lemma 3.1.2, ∈ Cl δ . By (3.1.8), for any ′ ∈ Cl δ , F ′ appears in F · F if and only if there is ′ ∈ r −1 δ ( ) such that ′ = ′ . This implies that r δ ( ′ ) = and, by definition of the δ-reduction map, for any ∈ [| |], ′ . Moreover, since ′ is a δ-cliff, we have for any . This is equivalent to the fact that ′ and leads to the expression of the statement of theorem.
For instance, for δ := 01120 ω , F 01 · F 010 = F 01010 + F 01020 + F 01110 + F 01120 and, since 01 011 = 01011 / ∈ Cl δ , F 01 · F 011 = 0 In particular when δ is weakly increasing, Lemma 3.1.3 and Theorem 3.1.4 state that any product of two elements of the F-basis of Cl δ is a sum of elements of the F-basis ranging in an interval of a δ-cliff poset.
3.2. Bases and algebraic study. We construct alternative bases for Cl δ and establish several properties of this structure w.r.t. properties of the range map δ. The main results are summarized in Table 1. We also provide a classification of all associative algebras Cl δ in four We now consider that δ is a weakly increasing range map. We need, to state the next result, to introduce for any 1 2 0 the two binary operations ⊣ ⊢ : Cl δ ( 1 ) × Cl δ ( 2 ) → N 1 + 2 defined, for any ∈ Cl δ , by ⊣ := ′ where ′ is the word on N of length | | satisfying, Notice that, by Lemmas 3. 2.1, 3.1.2, and 3.1.3, if δ is weakly increasing and and are δ-cliffs, both ⊣ and ⊢ are δ-cliffs.
Proposition 3.2.2. For any weakly increasing range map δ, the product · of Cl δ satisfies, for any ∈ Cl δ , Proof. By Lemma 3.2.1, we have The equality between the third and the last member of (3.2.13) is a consequence of the fact that for any ′′ ∈ N * , one has The equality between the third and the last member of (3.2.15) is a consequence of the fact that for any ′′ ∈ N * , one has r δ ( ′′ ) if and only if for all ∈ [| |], < δ( ) implies ′′ . By definition of the H-basis provided by (3.2.9), and since F r δ ( ) is the element with the greatest index appearing in the last member of (3.2.15), this expression is equal to the stated formula.

Presentation by generators and relations. A nonempty
The graded collection of all these elements is denoted by δ . For instance, for δ := 021 ω , among others, the δ-cliffs 0, 01, and 021111 are δ-prime, and 0210 = 021 0 and 011101 = 0111 01 are not.
Proposition 3.2.5. For any range map δ, the set {E : ∈ δ } is a minimal generating family of the unital magmatic algebra (Cl δ · 1).
Proof. Let us call G the set of the elements of Cl δ appearing in the statement of the proposition. We proceed by proving that any E , ∈ Cl δ , can be expressed as a product of elements of G by induction on the size of . Since for any λ ∈ K, 1(λ) = λE , E is the unit of Cl δ .
It remains to prove that G is minimal w.r.t. set inclusion. For this, let any ∈ δ and set G ′ := G \ {E }. Since by definition of δ-prime elements, and due to the product rule on the E-basis provided by Proposition 3.2.3, E cannot be expressed as a product of elements of G ′ , E is not generated by G ′ . Therefore, G is minimal.
Lemma 3.2.6. Let δ be a range map. If is a nonempty δ-cliff, then admits as suffix a unique δ-prime δ-cliff.
Proof. Assume by contradiction that there are two different suffixes and ′ of which are δ-prime. Therefore, | | = | ′ |, and the shortest word among and ′ is the suffix of the other. Assume without loss of generality that ′ is shorter than . This implies that there is a nonempty word ∈ N * such that = ′ . Now, since by hypothesis is a δ-cliff, and since any prefix of a δ-cliff is a δ-cliff, we obtain that is a δ-cliff. Therefore, = ′ where and ′ are both nonempty δ-cliffs. This implies that is not δ-prime, which is in contradiction with our assumptions.
Let A δ be the alphabet { : ∈ δ }. We denote by K A δ the free associative algebra generated by A δ . By definition, the elements of this algebra are noncommutative polynomials on A δ . For any ∈ Cl δ , we denote by the monomial (1) ( ) where (1) ( ) , 0, is the unique sequence of δ-prime δ-cliffs such that = (1) · · · ( ) . By Lemma 3.2.6, this definition is consistent since any δ-cliff admits exactly one factorization on δ-prime δ-cliffs. Proof. We use here the fact that, by Proposition 3.2.7, Cl δ is isomorphic as a unital associative algebra to the quotient K A δ / δ and the description of the generating familly of δ provided by Proposition 3.2.8.
Assume that δ is 1-dominated. The δ-cliffs 0, 1, . . . , δ(1) are δ-prime. Moreover, since δ is 1-dominated, there is an ℓ 0 such that any δ-cliff of size ℓ or more decomposes as = such that ∈ Cl δ (ℓ) and is a δ-cliff having only letters nongreater than δ(1). This implies that all δ-prime δ-cliffs have ℓ as maximal size. Therefore, Cl δ is finitely generated. Moreover, the finite number of nontrivial relations in Cl δ is the consequence of the finiteness of the generating set of Cl δ and the description of the relations of δ . Indeed, there is a finite number of monomials with ∈ Cl δ , ∈ δ , and / ∈ Cl δ that are not suffixes of any other one satisfying the same description. Conversely, assume that δ is not 1-dominated. Thus, since δ is valley-free, there is an index 1 such that δ(1) = · · · = δ( ) and for all + 1, δ( ) > δ(1). For any + 1, set as the δ-cliff of size defined by By Lemma 3.2.6, there is a unique δ-prime δ-cliff ′ being a suffix of . Since ′ is in particular a δ-cliff, one must have ′ 1 δ(1). Due to the previous description of δ, we necessarily have = ′ . Therefore, is δ-prime. This shows that there are infinitely many δ-prime δ-cliffs and thus, that Cl δ admits an infinite number of generators.
The set of all valley-free range maps can be partitioned into the following four classes: ⋆ The class of type A range maps, containing all constant range maps; ⋆ The class of type B range maps, containing all weakly increasing range maps having at least one ascent; ⋆ The class of type C range maps, containing all 1-dominated range maps having at least one descent; ⋆ The class of type D range maps, containing all range maps that are not 1-dominated and having at least one descent.
Theorem 3.2.11. Let δ be a valley-free range map. Each unital associative algebra (Cl δ · 1) admits the presentation K A δ / δ which fits into one of the following four classes: (i) If δ is of type A, then A δ is finite and δ is the zero space; (ii) If δ is of type B, then A δ is infinite and δ is the zero space; (iii) If δ is of type C, then A δ is finite and δ is finitely generated and nonzero; (iv) If δ is of type D, then A δ is infinite and δ is infinitely generated.
Proof. This is a consequence of the presentation by generators and relations of Cl δ provided by Propositions 3.2.7 and 3.2.8, and of the properties of the generating sets and relations spaces of Cl δ raised by Lemmas 3.2.9 and 3.2.10.
3.2.4. Examples. We provide here some examples of unital associative algebras Cl δ for particular range maps δ and describe their structure thanks to the classification provided by Theorem 3.2.11.
Type A. Let δ by a range map of type A. Thus, there is a value ∈ N such that δ( ) = for all ∈ N. Thus, Cl δ is the free unital associative algebra generated by 0 , 1 , . . . , .  A consequence of the freeness of Cl 1 is that Cl 1 is isomorphic as a unital associative algebra to FQSym [MR95,DHT02], an associative algebra on the linear span of all permutations. This follows from the fact that FQSym is also free as a unital associative algebra and that its Hilbert series is the same as the one of Cl 1 . Moreover, in [NT20], the authors construct some associative algebras FQSym as generalizations of FQSym whose bases are indexed by objects being generalizations of permutations. The algebras Cl m , 0, can therefore be seen as other generalizations of FQSym, not isomorphic to FQSym when 2.
Type C. Let the range map δ := 010 ω of type C. The unital associative algebra Cl δ admits the presentation Cl δ ≃ K 0 01 / δ where δ is minimally generated by the elements Let the range map δ := 210 ω of type C. The unital associative algebra Cl δ admits the presentation Cl δ ≃ K 0 1 2 / δ where δ is minimally generated (3.2.27) 3.3. Quotient algebras. This last section of this work provides an answer to the problem set out in the introduction. This question concerns the possibility of constructing a hierarchy of substructures of Cl δ similar to that of FQSym. For this, we consider quotients of Cl δ obtained by considering a graded subset of Cl δ and by equating the basis elements F with 0 whenever / ∈ . As we shall see, this is possible only under some combinatorial conditions on . We describe the products of these quotient algebras and give a sufficient condition for the fact that it can be expressed by interval of the poset ( ) for a certain 0. We end this part by studying the quotients of Cl m obtained from m-hills and m-canyons.
3.3.1. Quotient space. Let δ be a range map. Given a graded subset of Cl δ , let Cl be the quotient space of Cl δ defined by Cl := Cl δ / such that is the linear span of the set {F : ∈ Cl δ \ } By definition, the set {F : ∈ } is a basis of Cl .
Let us introduce here an important combinatorial condition for the sequel on . We say that is closed by suffix reduction if for any ∈ , for all suffixes ′ of , r δ ( ′ ) ∈ . Proposition 3.3.1. Let δ be a valley-free range map and be a graded subset of Cl δ . If is closed by prefix and is closed by suffix reduction, then Cl is a quotient algebra of the unital associative algebra (Cl δ · 1).
Proof. Notice first that, since δ is valley-free, Cl δ is by Theorem 3.1.1 a well-defined unital associative algebra. We have to prove that is an associative algebra ideal of Cl . For this, let F ∈ and F ∈ Cl . Let us look at Expression (3.1.8) for computing the product of Cl δ . Assume that there is a cliff ′ ∈ such that F ′ appears in F · F . Then, since is closed by prefix, ∈ , which contradicts our hypothesis. For this reason, F · F belongs to . Moreover, let F ∈ Cl and F ∈ . Assume that there is a cliff ′ ∈ such that F ′ appears in F · F . Then, since is closed by suffix reduction, one has r δ ( ′ ) ∈ . By (3.1.8), r δ ( ′ ) = , leading to the fact that ∈ holds, and which contradicts our hypothesis. Therefore, F · F belongs to . This establishes the statement of the proposition.
Notice that the graded subset Av δ is not closed by suffix reduction. For instance, even if 00112 is an 1-avalanche, the 1-reduction of its suffix 112 is 012, which is not an 1-avalanche.
Let us denote by θ : Cl δ → Cl the canonical projection map. By definition, this map satisfies, for any ∈ Cl δ , θ (F ) =↿ ∈ F 3.3.2. Product. We show here that under some conditions of , the product in Cl can be described by using the poset structure of . More precisely, we say that Cl has the interval condition if the support of any product F · F , ∈ , is empty or is an interval of a poset ( ), 0.

Lemma 3.3.2.
Let δ be a range map and be a graded subset of Cl δ such that for any 0, ( ) is a meet (resp. join) semi-sublattice of Cl δ ( ). For any ∈ , if is a δ-cliff, then the set [ ] ∩ admits at most one minimal (resp. maximal) element.
Proof. Assume that ( ) is a meet semi-sublattice of Cl δ ( ) and that ∈ Cl δ . By Lemma 3.1.2, ∈ Cl δ so that I := [ ] is a well-defined interval of Cl δ ( ). Assume that there exist two δ-cliffs and ′ belonging to I ∩ . Since ( ) is a meet semi-sublattice of Cl δ ( ), by setting ′′ := ∧ ′ , one has ′′ ∈ . Since is a lower bound of both and ′ , we necessarily have ′′ and ′′ ∈ I. This shows that when I ∩ is nonempty, this set admits exactly one minimal element. The proof is analogous for the respective part of the statement of the proposition.
When for any 0, ( ) is a lattice, we denote by ∧ (resp. ∨ ) its meet (resp. join) operation. In this case, is meet-stable (resp. join-stable) if, for any 0 and any ∈ ( ), the relation = for an ∈ [ ] implies that the -th letter of ∧ (resp. ∨ ) is equal to . . This, the definition of the ∨ operation, and the fact that is join-stable imply that ′′ . Therefore, ′′ ∈ I ∩ . This shows that when I ∩ is nonempty, this set admits exactly one maximal element. Proof. First, by Proposition 3.3.1, Cl is a well-defined unital associative algebra quotient of Cl δ . Now, the product F · F in Cl can be computed as the image by θ of the product of the same inputs in Cl δ . By Theorem 3.1.4, this product is equal to zero or its support I is an interval of a δ-cliff poset. By construction of Cl , the support of the product F · F in Cl is equal to I ′ := I ∩ . If (i) holds, then by Lemma 3.3.2, I ′ admits both a minimal and a maximal element. If (ii) holds, then by Lemma 3.3.2, I ′ admits a minimal element, and by Lemma 3.3.3, S ′ admits a maximal element. In both cases, I ′ is an interval of a poset ( ), 0.

3.3.3.
Examples: two Fuss-Catalan associative algebras. We define and study the associative algebras related to the m-hill posets and to the m-canyon posets.
Hill associative algebras. For any 0, let Hi be the quotient Cl Hi m . This quotient is well- We can observe that for any 1, Hi is not free as unital associative algebra. Indeed, the quasi-inverse of the respective generating series of these elements is not the Hilbert series of Hi , which is expected when this algebra is free. The associative algebra Ca 1 is the Loday-Ronco algebra [LR98], also known as PBT [HNT05]. It is known that this associative algebra is free and that the dimension of its generators are a shifted version of Catalan numbers: 0 1 1 2 5 14 42 132 429 (3.3.11) The sequence for the numbers of generators of Ca 2 degree by degree begins by 0 1 2 7 30 149 788 4332 (3.3.12) We can observe that for any 2, Ca is not free as unital associative algebra. It follows, from the same argument as the previous section, that Ca is not free.

C
This work presents three new families of posets on Fuss-Catalan objects and associative algebras on their linear spans. All this are based upon δ-cliffs, a combinatorial family of words of integers satisfying some conditions. Some general properties about subposets of the posets of δ-cliffs have been presented, as well as general properties about quotients of the associative algebras defined on the linear span of δ-cliffs.
Here is a list of open questions raised by this research: (1) (Generalization of the weak Bruhat order) -The first open question concerns the alternative order relation on δ-cliffs introduced in Section 1.2.3. This consists in considering Conjecture 1.2.2 and in proving that the posets (Cl δ ( ) ′ ) are semi-distributive lattices, or at least lattices.
(2) (Coproducts and Hopf bialgebras) -As explained above, the associative algebras Cl 1 and Ca 1 are already known algebraic structures which are in fact Hopf bialgebras. They are endowed with a coproduct satisfying some compatibility relations with the product. The question here consists in endowing Cl δ with a coproduct where δ is a unimodal range map. We can ask also for a general definition of such a coproduct for the quotients Cl of Cl δ for some subfamilies of δ-cliffs.
(3) (Other subposets and quotient algebras) -There are other subfamilies of δ-cliffs than δhills and δ-canyons which seem to lead to interesting posets and associative algebras. Among these, there are δ-dunes, which are δ-cliffs such that | − +1 | |δ( ) − δ( + 1)| for all ∈ [| | − 1]. For δ = 1, we obtain a family in one-to-one correspondence with directed animals, which are enumerated by Sequence A005773 of [Slo]. For δ = 2, we obtain a family enumerated by Sequence A180898 of [Slo]. This axis consists in studying in particular the posets and the associative algebras of dunes.