Pierre Catoire
email: pierre.catoire@univ-littoral.fr

Tridendriform Structures

Keywords: Hopf algebras, tridendriform, dendriform, Schröder trees 2020 Mathematics Subject Classification: 16S10, 16T05, 16T10, 16W50, 17A30

Inspired by the work of J-L. Loday and M. Ronco, we build free tridendriform algebras over reduced trees and we show that they have a coproduct satisfying some compatibilities with the tridendriform products. Its graded dual is the opposite bialgebra of TSym introduced by N. Bergeron et al., which is described by the lightening splitting of a tree. In particular, we can split the product in three pieces and the coproduct in two pieces with Hopf compatibilities. We generate its codendriform primitives and count its coassociative primitives thanks to L. Foissy's work.

Introduction

In particular cases, the associative algebra structure on a space A may be extended to a richer one. For example, in [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF] and [START_REF] Ronco | Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras[END_REF], the authors consider objects called dendriform algebras. In [START_REF] Loday | Scindement d'associativité et algèbres de Hopf[END_REF], they consider dendriform algebras and dipterous algebras. Studying such properties over the product of a bialgebra can give us information about its coproduct, if some convenient compatibilities are satisfied. We can also consider the dendriform or tridendriform cases (commutative or not) detailed in [START_REF] Foissy | Lie theory for quasi-shuffle bialgebras[END_REF] under the terminology of shuffle and quasi-shuffle algebras.

To be more precise, an algebra is dendriform if its product can be "broken" in two pieces denoted ≺, ≻ called respectively "left product" and "right product" such that these products verify, for all a, b, c ∈ A

(a ≺ b) ≺ c = a ≺ (b * c), (a ≻ b) ≺ c = a ≻ (b ≺ c), a ≻ (b ≻ c) = (a * b) ≻ c.
In particular, this implies that * :=≺ + ≻ is associative. The article [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF] adds the (dual) notion of dendriform coalgebra and describe dendriform bialgebras. In this document, we will bring our attention to the case tridendriform detailed in [START_REF] Loday | Trialgebras and families of polytopes[END_REF] and [START_REF] Burgunder | Tridendriform structure on combinatorial Hopf algebras[END_REF] with λ = 1. This means that our product * is split in three pieces denoted ≺, •, ≻, respectively called "left product", "middle product" and "right product" satisfying the properties of Definition 2.1. We define the associative product of a tridendriform algebra as * :=≺ + • + ≻. To study a tridendriform bialgebra H, it is convenient to define a tridendriform structure over H ⊗ H. In order to talk about coassociativity later, we remind that H ⊗ (H ⊗ H) and (H ⊗ H) ⊗ H have the same tridendriform structure Lemma 2.4. For any tridendriform algebras A, B and C, we have A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C as tridendriform algebras.

To build tridendriform bialgebras, we will need to add units. The unique way to add units is described in Section 2.2, with an extension ⊗ of the tensor product. We get Lemma 2.12. For A, B and C three augmented tridendriform algebras, we have A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C as tridendriform algebras.

After those reminders about the tensor products of tridendriform algebras and augmented tridendriform algebras, we will study the combinatorics of these algebras. For this we will use [START_REF] Loday | Trialgebras and families of polytopes[END_REF] where the free tridendriform algebra with one generator is described with reduced trees, i.e., trees where each node has at least 2 sons (also known as Schröder trees). This algebra is denoted by A. In this paper, we will get a new non-inductive formula for the tridendriform operations. Here are some examples of computations:

≺ = , ≺ = + + + + , • = , • = + + , ≻ = , ≻ = + + .
We will see in Section 4 that these products are quasi-shuffles of the branches of the right comb representation of the left term with the branches of the left comb representation of the right term. Looking at t as a right comb and s as a left comb means we consider them as

t = F 1 F 2 • • • F k and s = F k+1 F k+2 • • • F k+l
Definition 3.29. Let t and s be two trees which respective right comb and left comb representations are given above. Let k, l be the numbers of forests in the comb representation of t and s. Let σ be a (k, l)-quasi-shuffle which has for image 1, n . We denote σ(t, s) the tree obtained the following way:

1. We first consider the ladder with n nodes Node 1 Node 2 Node n 2. For all i ∈ 1, k , we graft F i as the left son at the node σ(i).

3. For all i ∈ k + 1, k + l , we graft F i as right son at the node σ(i). σ(t, s).

We also get a similar description for ≺, •, ≻ products. For more details, see Corollary 3.37. Then considering the number of leaves minus one as a gradation, we obtain Theorem 3.20. The unique coproduct ∆ over A compatible with the tridendriform structure and making (A, * , |, ∆, ε) a connected graded bialgebra is given by the following formula for all tree t

∆(t) = c admissible cut of t G c (t) ⊗ R c (t), ∆(|) = | ⊗ |,
where R c (t) is the component which has the root of t and G c (t

) = G c 1 (t) * • • • * G c k (t)
, where G c i (t) are naturally ordered. If c is the empty cut, we define R c (t) := t and G c (t) := |. If c is the total cut, we define R c (t) := | and G c (t) = t.

We shall use the notation (n, m)-dendriform bialgebras for n, m two non-negative integers, to talk about bialgebras where the product can be split in n other operations and the coproduct can be broken in m other operations, satisfying some Hopf compatibilities. Here is a table of known (n, m)-dendriform algebras, where n is the line number and m the column's one, In this paper, we introduce the new definitions of (3, 2) and (2, 3)-dendriform bialgebras. The reader will find some pieces of information about tridendriform algebras, cotridendriform coalgebras and cotridendriform tridendriform bialgebras in [START_REF] Burgunder | Structure theorems for dendriform and tridendriform algebras[END_REF] at Sections 5.1, 5.3 and 5.16, respectively. Tridendriform algebras are studied in [START_REF] Burgunder | Tridendriform structure on combinatorial Hopf algebras[END_REF]. Bidendriform bialgebras are introduced and studied in [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF]. The fourth section introduces the notion of (1, 3)-dendriform bialgebras which is the dual notion of (3, 1)-dendriform bialgebras. Thanks to the previous theorem, we give a link between the graded dual of A, denoted by A ⊛ , and the recent paper of N. Bergeron and al [START_REF] Bergeron | Hopf algebras of parking functions and decorated planar trees[END_REF], where the Hopf algebra TSym is introduced Theorem 4.23. The bialgebras A ⊛ and TSym op are the same.

So, TSym is a (1, 3)-dendriform bialgebra and we get a description of its graded dual. Finally, in the last section, we introduce the definition of (3, 2)-dendriform bialgebras

P. Catoire Definition 5.1. A (3, 2)-dendriform bialgebra is a sextuple (A, ≺, •, ≻, ∆ ← , ∆ → ) such that • (A, ≺, •, ≻) is an augmented tridendriform algebra. • (A, ∆ ← , ∆ → ) is an augmented dendriform coalgebra.
• Using generalized Sweedler's notations, the following relations are true, for all a, b ∈ A

∆ ← (a • b) = a ′ b ′ ← ⊗ a ′′ • b ′′ ← + b ′ ← ⊗ a • b ′′ ← , ∆ → (a • b) = a ′ b ′ → ⊗ a ′′ • b ′′ → + a ′ ⊗ a ′′ • b + b ′ → ⊗ a • b ′′ → , ∆ ← (a ≺ b) = a ′ b ′ ← ⊗ a ′′ ≺ b ′′ ← + a ′ b ⊗ a ′′ + b ′ ← ⊗ a ≺ b ′′ ← + b ⊗ a, ∆ → (a ≺ b) = a ′ b ′ → ⊗ a ′′ ≺ b ′′ → + a ′ ⊗ a ′′ ≺ b + b ′ → ⊗ a ≺ b ′′ → , ∆ → (a ≻ b) = a ′ b ′ → ⊗ a ′′ ≻ b ′′ → + a ′ ⊗ a ′′ ≻ b + b ′ → ⊗ a ≻ b ′′ → + ab ′ → ⊗ b ′′ → + a ⊗ b, ∆ ← (a ≻ b) = a ′ b ′ ← ⊗ a ′′ ≻ b ′′ ← + a ′ ⊗ a ′′ ≻ b + ab ′ ← ⊗ b ′′ ← .
We give an example of such an object.

Proposition 5.6. We consider (A, ≺, •, ≻) with its tridendriform algebra structure, we build the following coproducts:

∆← (t) =     c admissible cut of t the right leaf of t has been cut G c (t) ⊗ R c (t)     -t ⊗ 1, ∆→ (t) =     c admissible cut of t the right leaf of t has not been cut G c (t) ⊗ R c (t)     -1 ⊗ t. Then (A, ≺ +•, ≻, ∆ ← , ∆ → ) and (A, ≺, •+ ≻, ∆ ← , ∆ → ) are bidendriform bialgebras. As a con- sequence, (A, ≺, •, ≻, ∆ ← , ∆ → ) is a (3, 2)-dendriform bialgebra.
There, we can apply the results from [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF] to count and describe the codendriform primitives ans coassociative primitives. With this work, we are able to get all the codendriform primitives and we can count the coassociative ones. Finally, to end the paper, we study the "natural" quotient of A as a (3, 2)-dendriform algebra by the set of elements obtained with the middle product • (we send the middle product • to 0) in order to get back to the Loday-Ronco algebra given in [START_REF] Loday | Hopf algebra of the planar binary trees[END_REF] and [START_REF] Aguiar | Structure of the Loday-Ronco Hopf algebra of trees[END_REF].

Some details about the bibliography

For other examples of tridendriform algebras, we can refer to the quasi-shuffle algebras present in [START_REF] Loday | On the algebra of quasi-shuffles[END_REF][START_REF] Novelli | Binary shuffle bases for quasi-symmetric functions[END_REF] or [START_REF] Aguiar | Combinatorial Hopf algebras and generalized Dehn-Sommerville relations[END_REF]. The tridendriform algebra of the parking functions is given in [START_REF] Burgunder | Tridendriform structure on combinatorial Hopf algebras[END_REF]. In the document, Remarks 3.2, 3.19 and 3.31 describe the case with multiple generators. They are given without proofs, as it follows from the proofs detailed for one generator.

Document structure

In all the document, K will be any commutative field. This document has four sections:

• The first one gives several reminders about tridendriform algebras and its augmentations.

• The following one describes the free tridendriform bialgebra (relying on [START_REF] Loday | Trialgebras and families of polytopes[END_REF]) with one generator, giving combinatorial descriptions of the product with quasi-shuffle and admissible cuts. There is also a description for multiple generators given as remarks.

• Then we study the graded dual of the free tridendriform bialgebra which is TSym, the bialgebra whose coproduct is given by the lightning decomposition of a tree. As a consequence of this section, we will find a (1, 3)-dendriform bialgebra structure for TSym.

• The last section defines (3, 2)-dendriform bialgebras and describe one example built on the free (3, 1)-dendriform bialgebra with one generator given in the third section. We get it by decorating the right-most leaf of a tree and we check it satisfies the definition. Finally, we will show that a quotient of the free (3, 2)-dendriform algebra is the Loday-Ronco algebra.

Reminder

2.1 Tensor product for tridendriform algebras Definition 2.1. Let K be a field and A be vector space of K. We say (A, ≺, •, ≻) is a tridendriform algebra if ≺, • and ≻ are all linear maps from A ⊗ A to A such that for any a, b, c ∈ A

(a ≺ b) ≺ c = a ≺ (b * c), (2.1)
(a ≻ b) ≺ c = a ≻ (b ≺ c), (2.2)
(a * b) ≻ c = a ≻ (b ≻ c), (2.3)
(a ≻ b) • c = a ≻ (b • c), (2.4)
(a ≺ b) • c = a • (b ≻ c), (2.5)
(a • b) ≺ c = a • (b ≺ c), (2.6)
(a • b) • c = a • (b • c), (2.7)
where * is the associative product of the tridendriform algebra defined for any a, b ∈ A by

a * b := a ≺ b + a • b + a ≻ b.
We call the products ≻, ≺, • respectively right, left and middle.

Remark 2.2. The associativity of * is a consequence of those seven relations.

Let us consider an associative algebra (A, ⋆) and a tridendriform algebra (B, ≻, •, ≺). Our goal is to give A ⊗ B a tridendriform algebra structure. In order to reach this aim, let us define for all ⋉ ∈ { * , ≺, ≻, •} ⋉ :

(A ⊗ B) ⊗ (A ⊗ B) → A ⊗ B, (a ⊗ b) ⊗ (c ⊗ d) → a ⋆ c ⊗ b ⋉ d, With these definitions, we have * =≺ + • + ≻. Lemma 2.3. (A ⊗ B, ≺, •, ≻) is a tridendriform structure.
Proof . We wish that these definitions make A ⊗ B a tridendriform algebra. We will check the relations (2.1) to (2.7). To shorten the proof, we notice that the relations can be written as follows for all a, b, c ∈ A

(a ⋉ b) ⋊ c = a ⋉ ′ (b ⋊ ′ c),
for (⋉, ⋊, ⋉ ′ , ⋊ ′ ) belonging to the following set:

{(≺, ≺, ≺, * ), (≻, ≺, ≻, ≺), ( * , ≻, ≻, ≻), (≻, •, ≻, •), (≺, •, •, ≻), (•, ≺, •, ≺), (•, •, •, •)}.
(2.8)

Let (⋉, ⋊, ⋉ ′ , ⋊ ′ ) be an element of the set above. Let a, c, e ∈ A and b, d, f ∈ B, then

((a ⊗ b) ⋉ (c ⊗ d)) ⋊ e ⊗ f = (a ⋆ c ⊗ b ⋉ d) ⋊ (e ⊗ f ) = (a ⋆ c) ⋆ e ⊗ (b ⋉ d) ⋊ f = a ⋆ (c ⋆ e) ⊗ b ⋉ ′ (d ⋊ ′ f ) = (a ⊗ b) ⋉ ′ ((c ⊗ d) ⋊ ′ (e ⊗ f )).
Consequently, this shows that A ⊗ B is a tridendriform algebra. ■

Moreover, the tensor product for tridendriform algebras is associative.

Lemma 2.4. For any tridendriform algebras A, B and C, we have A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C as tridendriform algebras.

Proof . One just needs to write down the definitions of the products. ■

Augmented tridendriform algebras

Let (A, ≻, ≺, •) be a tridendriform algebra. If we do some easy computations, we find that (A, ≻ +•, ≺) and (A, ≻, ≺ +•) are two dendriform algebras, as detailed in [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF]. In order to define consistently the concept of augmented tridendriform algebra, which we will denote by (A, ≻, ≺, •) where A = A ⊕ K1 (1 is a unit for * that we formally add to A), we need that (A, ≻ +•, ≺) and (A, ≻, ≺ +•) are both augmented dendriform algebras as defined in [START_REF] Ebrahimi-Fard | Monotone, free, and boolean cumulants: A shuffle algebra approach[END_REF]Section 3.1]. Therefore, the definition of dendriform algebras applied to (A,

≻ +•, ≺) gives us for all a ∈ A a ≺ 1 + a • 1 = a = 1 ≻ a and 1 ≺ a + 1 • a = 0 = a ≻ 1.
Applying the same definition to (A, ≻, ≺ +•), we get for all a ∈ A a

≺ 1 = a = 1 ≻ a + 1 • a and 1 ≺ a = 0 = a ≻ 1 + a • 1.
As a consequence, we need to define 1 • a = a • 1 = 0 for all a ∈ A.

Definition 2.5. Let (A, ≻, ≺, •) be a tridendriform algebra. We expand the products on

K ⊗ A ⊕ A ⊗ K ⊕ A ⊗ A by defining ∀a ∈ A, 1 ≺ a := 0 = 1 ≻ a, a ≺ 1 := a = 1 ≻ a and a • 1 := 0 = 1 • a. (2.9)
This construction gives us what we will call augmented tridendriform algebra. For a tridendriform algebra A, we denote A the associated augmented tridendriform algebra.

Remark 2.6. With this definition of augmented tridendriform algebra, when both terms of the relations (2.1)-(2.7) are well defined, there is an equality.

Remark 2.7. This is the unique way to define a ≻ 1, 1 ≻ a, a ≺ 1 and 1 ≺ a such that (2.1)-(2.7) are equalities as soon as both sides are well defined.

Remark 2.8. It is not possible to define 1 ≺ 1 and 1 ≻ 1 such that the relations (2.9) are satisfied over A.

Tensor product for augmented tridendriform algebras

Let A, ≻, ≺, • and B, ≻, ≺, • be two augmented tridendriform algebras. As in the dendriform case, it is impossible to define ≻ and ≺ on the pair (1, 1) such that (2.9) still occurs. For this reason, we define

A ⊗ B := A ⊗ B ⊕ K ⊗ B ⊕ A ⊗ K.
The previous vector space is a subspace of A⊗B on which we will define the three operations ≻, ≺ and •.

Remark 2.9.

• We can extend * to A ⊗ A with 1 * 1 = 1. Moreover * is associative on A.

• For two tridendriform algebras A and B, we have

A ⊗ B = A ⊗ B.
We define * :

A ⊗ B ⊗2 → A ⊗ B , a ⊗ b ⊗ c ⊗ d → a * c ⊗ b * d.
To simplify the notations, we denote by

D = (A ⊗ K) ⊗2 ⊕ ((K ⊗ K) ⊗ (A ⊗ K)) ⊕ ((A ⊗ K) ⊗ (K ⊗ K)) , U = (A ⊗ B) ⊗2 ⊕ ((A ⊗ K) ⊗ (A ⊗ B)) ⊕ ((A ⊗ B) ⊗ (A ⊗ K)) .
So, (A⊗B) ⊗2 = U ⊕ D. Then for all ⋉ ∈ {≺, •, ≻}, we define

⋉ :      (A ⊗ B) ⊗2 → A ⊗ B , a ⊗ b ⊗ c ⊗ d ∈ U → a * c ⊗ b ⋉ d, a ⊗ b ⊗ c ⊗ d ∈ D → a ⋉ c ⊗ b * d.
Lemma 2.10. We consider two tridendriform algebras A and B. The algebra A ⊗ B is a tridendriform algebra with the products defined above.

Remark 2.11. This implies that A ⊗ B is an augmented tridendriform algebra.

Proof . In Section 2.1, we have checked that the relations (2.1)-(2.7) are true when a, b, c, d, e, f are elements of A or B. We now need to check those relations when at least one of these elements is a unit. Let (⋉, ⋊, ⋉ ′ , ⋊ ′ ) be an element of the set defined at (2.8). To simplify the following, when we will consider x ∈ K1, we will suppose x = 1 after multiplying by a scalar. There are four cases to check:

• Case 1: d ∈ B or (b ∈ B and f ∈ B); • Case 2: b, d, f ∈ 1 B K; • Case 3: d ∈ 1 B K, f ∈ 1 B K and b ∈ B; • Case 4: b ∈ 1 B K, d ∈ 1 B K, f ∈ B.
We prove Case 3 as an example. Other cases are left to the reader. We suppose d ∈ 1 B K, f ∈ 1 B K and b ∈ B. Then c and e cannot be units.

[(a ⊗ b) ⋉ (c ⊗ 1)] ⋊ (e ⊗ 1) = (a * c ⊗ b ⋉ 1) ⋊ (e ⊗ 1) = (a * c) * e ⊗ (b ⋉ 1) ⋊ 1 = a * (c * e) ⊗ b × δ (⋉,⋊),(≺,≺) , (a ⊗ b) ⋉ ′ [(c ⊗ 1) ⋊ ′ (e ⊗ 1)] = a ⊗ b ⋉ ′ (c ⋊ ′ e ⊗ 1) = a * (c ⋊ ′ e) ⊗ b ⋉ ′ 1 = a * (c ⋊ ′ e) ⊗ b ⋉ ′ 1 × δ (⋉ ′ ,⋊ ′ ),(≺, * ) .
Having a look at (2.8), we see that the only quadruple with (⋉, ⋊) = (≺, ≺) is (≺, ≺, ≺, * ). So, there is equality in this case. Once, we have checked the relations (2.1)-(2.7) on A ⊗ B. There is still to check * =≺ + • + ≻. With the work done in the first section, we only need to compute this when there are at least one unit. Those verifications are straightforward. ■ Proof . In the following, we will use the identifications

Associativity of ⊗

A ⊗ K ≈ A, B ⊗ K ≈ B, C ⊗ K ≈ C and A ⊗ (B ⊗ C) ≈ (A ⊗ B) ⊗ C.
As a consequence, we see that A ⊗ B ⊗ C and A ⊗ B ⊗ C have the same underlying vector space. We put D the following set:

A ⊗ B ⊗ C ⊗2 ⊕ A ⊗ B ⊗ C ⊗ A ⊗ B ⊗ K ⊕ A ⊗ B ⊗ K ⊗ A ⊗ B ⊗ C .
For any ⋉ ∈ { * , ≺, ≻, •}, the tridendriform structure over ((A ⊗ B) ⊗ C) ⊗2 is given by

⋉ 1 :      ((A ⊗ B) ⊗ C) ⊗2 → (A ⊗ B) ⊗ C, (a ⊗ b) ⊗ c ⊗ (a ′ ⊗ b ′ ) ⊗ c ′ ∈ D → (a ⊗ b) * (a ′ ⊗ b ′ ) ⊗ c ⋉ c ′ , (a ⊗ b) ⊗ c ⊗ (a ′ ⊗ b ′ ) ⊗ c ′ ∈ ((A⊗B) ⊗ K) ⊗2 → (a ⊗ b) ⋉ (a ′ ⊗ b ′ ) ⊗ c * c ′ .
Extending by linearity, we get an map ⋉ 1 defined on all ((A ⊗ B) ⊗ C) ⊗2 . Let us put D ′ the set

A ⊗ (B ⊗ C) ⊗2 ⊕ A ⊗ K ⊗2 ⊗ A ⊗ (B ⊗ C) ⊕ A ⊗ (B ⊗ C) ⊗ A ⊗ K ⊗2 .
The tridendriform structure on A ⊗ B ⊗ C is given for all ⋉ ∈ { * , ≺, ≻, •} by

⋉ 2 :      (A ⊗ (B ⊗ C)) ⊗2 → A ⊗ (B ⊗ C), a ⊗ (b ⊗ c) ⊗ a ′ ⊗ (b ′ ⊗ c ′ ) ∈ D ′ → a * a ′ ⊗ [(b ⊗ c) ⋉ (b ′ ⊗ c ′ )], a ⊗ (b ⊗ c) ⊗ a ′ ⊗ (b ′ ⊗ c ′ ) ∈ A ⊗ K ⊗2 ⊗2 → a ⋉ a ′ ⊗ [(b ⊗ c) * (b ′ ⊗ c ′ )].
By linearity of ⋉ 1 and ⋉ 2 , for all x ∈ A. Then doing some straightforward computations, we obtain for all ⋉ ∈ {≺, ≻, •, * }, ⋉ 1 = ⋉ 2 . This proves the lemma. ■ Notation 2.13. Let A be an augmented tridendriform algebra. We denote by A + the augmentation ideal of A.

Remark 2.14. With those notations, we have A + = A.

Before ending this section, we will need a definition and an easy lemma.

Definition 2.15. Let A, B be two tridendriform algebras. Let f : A → B a linear map. We say that f is a tridendriform algebra morphism if for all a, b ∈ A and for all ⋉ ∈ {≺, •, ≻}

f (a ⋉ b) = f (a) ⋉ f (b).
If A, B are augmented, we say that f is an augmented tridendriform algebra morphism if

f : A + → B + is a tridendriform algebra morphism with f (1) = 1.
Lemma 2.16. Let A, A ′ , B, B ′ be four augmented tridendriform algebras. Consider two tridendriform algebra morphisms f : A → A ′ and g : B → B ′ . Then the map

f ⊗g : A + ⊗ B + → A ′ ⊗ B ′ , a ⊗ b → f (a) ⊗ g(b)
is a tridendriform algebra morphism. Defining

f ⊗ g :      A ⊗ B → A ⊗ B, a ⊗ b ∈ A + ⊗ B + → (f ⊗ g)(a ⊗ b), 1 ⊗ 1 → 1 ⊗ 1,
we get an augmented tridendriform algebra morphism.

Proof . The proof is straightforward. ■

3 The free tridendriform algebra and its coproduct

Let Tridend be the quadratic operad generated by Tridend 2 = ⟨≻, ≺, •⟩ satisfying the relations (2.1)-(2.7). Let us consider A + = Tridend(K). In other words, A + is a free tridendriform algebra generated by one element. Referring to [15, Theorem 2.6], we get A + = n≥1 KT n where T n is the set of all planar rooted trees with n + 1 leaves in which every internal node has fertility at least 2. In other words, T n is the set of reduced trees (also called Schröder trees) with n + 1 leaves. Then we add a unit to this tridendriform algebra. We will denote in the following

A = A + , A = n≥0 KT n . Examples 3.1. T 0 = {|}, T 1 = { }, T 2 = , , . Remark 3.2.
• For more generators, we need to define: Let n ∈ N. Let D be a graded set by N, this means D = i∈N D i . We denote T g n (D) the set of all rooted trees with n + 1 leaves such that each internal node x is decorated by an element of D n where n is the number of children of the internal node x.

• Let X be a non-empty set. The vector space on which we define the free tridendriform algebra generated by X is

A(X) = +∞ n=0 T g n (X ⋆ ),
where X ⋆ is the set of all finite words on the alphabet X and for all n ∈ N, X ⋆ n is the set of all words of length n over the alphabet X. An equivalent definition exists decorating the angular sectors of a tree by one letter from X instead. For more details, refer to [21, Section 4]. For example, this is two representations of a same element of A(X): Let k be a non-negative integer. Let x (0) , . . . , x (k) be trees from

T n i for i ∈ 1, k and n i ∈ N. We denote by x (0) ∨ • • • ∨ x (k)
the tree obtained by grafting on the same root and from left to right the trees x (0) , . . . , x (k) . Then for x and y two trees, there exist unique trees x (0) , . . . , x (k) and y (0) , . . . , y (l) such that

x = x (0) ∨ • • • ∨ x (k) , y = y (0) ∨ • • • ∨ y (l) .
With those notations, Theorem 2.6 from [START_REF] Loday | Trialgebras and families of polytopes[END_REF] tells us that we can define the three products inductively by 2) is the following tree:

x ≺ y = x (0) ∨ • • • ∨ x (k) * y , x • y = x (0) ∨ • • • ∨ x (k) * y (0) ∨ • • • ∨ y (l) , x ≻ y = x * y (0) ∨ • • • ∨ y (l) , initializing the induction with | ≺ y = 0, y ≺ | = y, x ≺ | = x, x ≻ | = 0, | • y = 0 = | • | and | * | = |. Examples 3.3. • Taking k = 2 and x (0) = |, x (1) = , x (2) = |, x (0) ∨ x (1) ∨ x (
.

• On the other hand, the tree denoted by x (1) ∨ x (0) ∨ x (2) is .

Our investigation to find a bialgebra with a tridendriform structure begins here. With this objective in mind, we need to endow A with a bialgebra structure. Using the dendriform structure of A ⊗ A, it is natural to impose for all x, y ∈ A

∆(x ≺ y) = ∆(x) ≺ ∆(y), ∆(x • y) = ∆(x) • ∆(y), ∆(x ≻ y) = ∆(x) ≻ ∆(y),
which leads to the following definition:

Definition 3.4. We will call a tridendriform bialgebra or (3, 1)-dendriform bialgebra any augmented tridendriform algebra (H, ≻, ≺, •) which also has a bialgebra structure (H, * , 1, ∆, ε), where * =≺ + • + ≻, satisfying the following compatibilities, for all x, y ∈ H

+ ∆(x ≺ y) = ∆(x) ≺ ∆(y), ∆(x • y) = ∆(x) • ∆(y), ∆(x ≻ y) = ∆(x) ≻ ∆(y).
This notion of tridendriform bialgebra is such that (H, ≺, ≻ +•, 1 H , ∆, ε) and (H, ≺ +•, ≻, 1 H , ∆, ε) are both dendriform bialgebras, see Definition 1.3 with q = 1 from [START_REF] Burgunder | Tridendriform structure on combinatorial Hopf algebras[END_REF]. Let us remind these definitions. Definition 3.5. Let (H, m, 1, ∆, ε) be a bialgebra. We say H is graded if there exists a sequence (H n ) n∈N of vectorial subspaces of H of finite dimension verifying

H = +∞ n=0 H n .
Moreover, this grading satisfies

• ∀i, j ∈ N, m(H i ⊗ H j ) ⊆ H i+j , • ∀n ∈ N, ∆(H n ) ⊆ i+j=n H i ⊗ H j . Definition 3.6. A graded Hopf algebra H is called connected if H 0 has dimension 1.
Then giving A the grading A n = KT n for all n ∈ N, we also wish that (A, * , |, ∆, ε) is a graded and connected bialgebra. Notation 3.7. For any tree t, nl(t) is the number of leaves of t and deg(t) = nl(t) -1.

Coproduct description

In this section, we will look for a coalgebra structure on (A, ≺, •, ≻). Definition 3.8. We define the following tridendriform algebra morphism:

∆ :      A → A⊗A, → ⊗ | + | ⊗ , | → | ⊗ |.
But, as A is the free tridendriform algebra generated by one element, this morphism exists and is unique because A⊗A is also a tridendriform algebra as seen in the previous section. We will check at Theorem 3.20 that ∆ is coassociative. Notation 3.9. For any n ∈ N, we denote by c n the corolla with n + 1 leaves

• • • .
Remark 3.10. We describe here an explicit technique in order to find one writing of a tree.

Let n ∈ N and x ∈ T n . We write x uniquely as k) . We need to consider four cases:

x = x (0) ∨ • • • ∨ x (
• Case 1: ∀i ∈ 0, k , x (i) = |. Then x = • c k . • Case 2: x (0) ̸ = |. Then x = x (0) ≻ | ∨ x (1) ∨ • • • ∨ x (k-1) ∨ x (k) .
• Case 3:

x (k) ̸ = |. Then x = (x (0) ∨ • • • ∨ x (k-1) ∨ |) ≺ x (k) .
• Case 4: there exists i ∈ 1, k -1 such that x (i) ̸ = |. Then we obtain

x = x (0) ∨ • • • ∨ x (i-1) ∨ x (i) ∨ x (i+1) ∨ • • • ∨ x (k) = x (0) ∨ • • • ∨ x (i-1) ∨ | • x (i) ∨ x (i+1) ∨ • • • ∨ x (k) = x (0) ∨ • • • ∨ x (i) • | ∨ x (i+1) ∨ • • • ∨ x (k) ,
where the two writings are equal thanks to the relation (2.5). Then we apply this process once more on the branches of x and so on. Finally, we get a writing of x over the language {≺, •, ≻} with constant symbol .

In the following, we will give a lemma that helps us to find an explicit writing of ∆.

Notation 3.11. For all x ∈ A + , we denote ∆(x) = ∆(x) -|⊗ x -x ⊗|.

Lemma 3.12. Let t, s ∈ A = Tridend(K). Then

∆(t • s) = ∆(t) • ∆(s) + (1 ⊗ t) • ∆(s) + ∆(t) • (1 ⊗ s), ∆(t ≺ s) = s ⊗ t + (1 ⊗ t) ≺ ∆(s) + ∆(t) ≺ (1 ⊗ s) + ∆(t) * (s ⊗ 1) + ∆(t) ≺ ∆(s), ∆(t ≻ s) = t ⊗ s + (1 ⊗ t) ≻ ∆(s) + (t ⊗ 1) * ∆(s) + ∆(t) ≻ (1 ⊗ s) + ∆(t) ≻ ∆(s). Proof . Let ⋉ ∈ {≻, •, ≺}. Let t, s ∈ A. Then ∆(t ⋉ s) = 1 ⊗ t + t ⊗ 1 + ∆(t) ⋉ 1 ⊗ s + s ⊗ 1 + ∆(s) = 1 ⊗ t ⋉ s + s ⊗ t ⋉ 1 + (1 ⊗ t) ⋉ ∆(s) + t ⊗ 1 ⋉ s + t ⋉ s ⊗ 1 + (t ⊗ 1) ⋉ ∆(s) + ∆(t) ⋉ (a ⊗ s) + ∆(t) ⋉ (s ⊗ 1) + ∆(t) ⋉ ∆(s).
We get the lemma applying the definition of ≺, • and ≻ on the equation above. ■ Definition 3.13. Let t be a tree, we define the valence of a vertex v of t as the number of outgoing edges of v. A leaf of t is an edge of valence 0. We call a vertex of t internal if it is not a leaf. We say that an edge of t is internal if it links two internal vertices.

Definition 3.14. Let t be a tree. A cut of t is a non-empty choice of internal edges of t. We call empty cut of t, the cut which chooses no edge of t and we call total cut the cut "under the root of t".

A cut c of t is called admissible if all paths from the root to one of the leaves do not meet more than one chosen edge by c. The total and empty cuts are admissible cuts.

Example 3.15. On the tree the drawn cut is not admissible but is admissible.

Notation 3.16. Let t be a tree and c be an admissible cut of t which is neither empty, neither total. By withdrawing the chosen edges by c, we find some trees that we will denote G c 1 (t), . . . , G c m (t), where m is the number of edges chosen by c, G c 1 (t) is the left-most tree after withdrawing all edges of c, . . . , and G c m (t) is the right-most tree. We also denote by R c (t) the component after withdrawing the edges of c containing the root of t. If c is the empty cut, we define R c (t) := t and G c (t) := |. If c is the total cut, we define R c (t) := | and G c (t) = t.

Examples 3.17.

• The empty cut c of the tree t gives R c (t) = t.

• The total cut c of t gives G c (t) = t.

• Let us consider the tree with the cut c symbolized by the horizontal line cutting the tree. In this case, R c (t) = and G c (t) = .

• With , we get

R c (t) = and G c 1 (t) = , G c 2 (t) = , G c 3 (t) = .
Observing our results, we get Proposition 3.18. The map defined at Definition 3.8 over A is given for all trees t by the formula

∆(t) = c admissible cut of t G c (t) ⊗ R c (t), ∆(|) = | ⊗ |,
where R c (t) is the component of t containing its root and

G c (t) = G c 1 (t) * • • • * G c k (t)
, where G c i (t) are naturally ordered from left to right.

Proof . We proceed by induction over the number of leaves. As seen before,

∆ = ⊗ | + | ⊗ .
This corresponds to the statement of our proposition. Heredity: let n ∈ N. Suppose that for all trees with at most n + 1 leaves, their coproduct is written like in the proposition. Let t be a tree with n + 2 leaves. There exist k, s ∈ N such that k + s = n + 3 and u ∈ T k , v ∈ T s such that one of these equations is true:

t = u ≺ v, where u = u (0) ∨ u (1) ∨ • • • ∨ u (r-1) ∨ |, (3.1)
t = u ≻ v, where v = | ∨ v (1) ∨ • • • ∨ v (r-1) ∨ v (r) , (3.2)
t = u • v, where u = u (0) ∨ • • • ∨ u (r-1) ∨ |, (3.3)
using Remark 3.10 and r denotes the valence of the root of t. Under the condition (3.1), we have

∆(t) = ∆(u) ≺ ∆(v) = cu admissible cut of u G cu (u) ⊗ R cu (u) ≺ cv admissible cut of v G cv (v) ⊗ R cv (v) = cu admissible cut of u cv admissible cut of v R cu ̸ =|∧R cv ̸ =| R cu ̸ =u∧R cv ̸ =v G cu (u) * G cv (v) ⊗ R cu (u) ≺ R cv (v) + | ⊗ u ≺ v + u ≺ v ⊗ |. But u = u (0) ∨ u (1) ∨ • • • ∨ u (r-1) ∨ |. As a consequence, an admissible cut of u ≺ v = u (0) ∨ u (1) ∨ • • • ∨ u (r-1)
∨ v is an admissible cut of u followed by an admissible cut of v. Moreover, the branches falling from v fall at the right of the branches from u. This implies

∆(u ≺ v) = ∆(t) = c admissible cut of t G c (t) ⊗ R c (t).
With an analogous idea, we get the same result in the case (3.2). In the case (3.3), we get

t = u • v with u = u (0) ∨ • • • ∨ u (r-1) ∨ |. As a consequence, ∆(t) = cu admissible cut of u G cu (u) ⊗ R cu (u) • cv admissible cut of v G cv (v) ⊗ R cv (v) = | ⊗ u • v + u • v ⊗ | + cu admissible cut of u cu admissible cut of v R cu ̸ =|∧R cv ̸ =| R cu ̸ =u∧R cv ̸ =v G cu (u) * G cv (v) ⊗ R cu (u) • R cv (v). We denote v = v (0) ∨ • • • ∨ v (r ′ ) . With the shape of u, an admissible cut of u • v = u (0) ∨ • • • ∨ u (r-1) ∨ v (0) ∨ • • • ∨ v (r ′
) is an admissible cut of u followed by an admissible cut of v. Moreover, the branches of v fall at the right of the branches of u. This shows that

∆(t) = c admissible cut of t G c (t) ⊗ R c (t). ■ Remark 3.19.
In the case of multiple generators, the coproduct over A(D) is the same extending naturally its definition to decorated trees.

Theorem 3.20. The coproduct of Proposition 3.18 is coassociative and has for counit the linear map defined by

ε(t) = 1 if t = |, 0 else.
As a consequence, (A, * , |, ∆, ε) is a connected graded bialgebra and a tridendriform bialgebra. Moreover, ∆ is the unique map making (A, * , |) a tridendriform bialgebra.

Proof . It is enough to show that ∆ is coassociative, unique and ε satisfies the counit property. First, let us remind that A + is the free tridendriform algebra generated by one element. The result of Lemma 2.3 says that A + ⊗ A + is a tridendriform algebra. Using the universal property of the free tridendriform algebra given by the operad theory, we find the existence of a unique tridendriform algebra morphism such that ∆ :

A + → A + ⊗A + , → | ⊗ + ⊗ |.
Then we expand the definition of ∆ to A by defining ∆(|) = | ⊗ |. Consequently, we have the uniqueness of ∆. We get two maps

(Id ⊗ ∆) • ∆ : A → A ⊗ (A ⊗ A), (∆ ⊗ Id) • ∆ : A → (A ⊗ A) ⊗ A.
But we have seen that the final spaces of these two maps have the same augmented tridendriform structure in Lemma 2.12. By Lemma 2.16, (∆ ⊗ Id) • ∆ and (Id ⊗ ∆) • ∆ are morphisms of augmented tridendriform algebras. Then we deduce that these two applications are equal over

A + because (Id ⊗ ∆) • ∆ = (∆ ⊗ Id) • ∆ and generates A + as tridendriform algebra. Moreover, (Id ⊗ ∆) • ∆(|) = (∆ ⊗ Id) • ∆(|). So, (∆ ⊗ Id) • ∆ = (Id ⊗ ∆) • ∆ over A. So ∆ is coassociative.
For the counit property, using Lemma 2.16, we know that (ε ⊗ Id) • ∆ and (Id ⊗ ε) • ∆ are morphisms of augmented tridendriform algebras. Moreover,

(ε ⊗ Id) • ∆ = = (Id ⊗ ε) • ∆ .
Thanks to the universal property of the free tridendriform algebra A + , these two morphisms are equal to Id. It is clearly the same for ⟨|⟩. So, it is the identity over A. ■ Definition 3.21. For any bialgebra (H, * , 1, ∆, ε), we define

Prim(H) := x ∈ H | ∆(x) = 0 .
Corollary 3.22. Every tree t element of Prim(A) is a corolla.

Proof . Let t ∈ Prim(A). If t has two leaves, then t = . Let us suppose t has at least three leaves. Then using the fact that generates the tridendriform algebra A, we deduce the existence of two trees u, v which have strictly less leaves than t such that

t = u ≺ v or t = u • v or t = u ≻ v.
By Lemma 3.12, the only possibility for t to be primitive is t = u • v with u, v ∈ Prim(A). By induction hypothesis, u and v are corollas. So t is a corolla. ■ Examples 3.23.

• • = | ∨ (| * |) ∨ | = is primitive. • • = | ∨ | ∨ (| * |) ∨ | = is primitive.
3.2 Description of the product

Quasi-shuffles

In this part, our objective is to give non inductive descriptions of the product * and the three others operations ≺, •, ≻. We shall need the following definition:

Definition 3.24 (quasi-shuffle). Let k, l ∈ N \ {0}. A (k, l)-quasi-shuffle is a surjective map σ : 1, k + l ↠ 1, n surjective for some positive integer n such that σ(1) < • • • < σ(k) and σ(k + 1) < • • • < σ(k + l).
We will denote QSh(k, l) the set of all (k, l)-quasi-shuffles.

The reader may find more details about quasi-shuffles and tridendriform algebras in [START_REF] Palacios | Weak Bruhat order on the set of faces of the permutohedron and the associahedron[END_REF].

Remark 3.25. We will use the following notation to describe quasi-shuffles. Let k, l ∈ N \ {0}.

Let σ be a surjective map of 1, k + l into some 1, n . We will denote σ like this (σ(1), . . . , σ(k), σ(k + 1), . . . , σ(k + l)).

Remark 3.26. Let σ ∈ QSh(k, l), then σ -1 ({1}) is either {1}, {k + 1} or {1, k + 1}.

Comb representation of a tree

Let t be a tree. It t can always be seen as

t 1 1 t 1 n 1 • • • t 2 1 t 2 n 2 • • • • • • t k 1 t k n k • • •
where k is the number of nodes on the right-most branch of t and for i ∈ 1, k , n i + 1 is the number of sons of the i-th node of this branch.

Notation 3.27. Let F = t 1 . . . t n be a forest composed of n trees. Instead of writing

t 1 t n • • • we will write F .
As a consequence, we deduce that all tree t can be seen as

F 1 F 2 • • • F k
where F 1 , . . . , F k are non-empty forests.

Definition 3.28. The writing defined above is the writing of t as a right comb. Proceeding the same way, looking at the left branch of the root of t, we get the writing of t as a left comb.

Action of the quasi-shuffles over ordered pairs of trees

Let t, s two trees different from |. We look t as a right comb and s as a left comb. In other words, we put

t = F 1 F 2 • • • F k and s = F k+1 F k+2 • • • F k+l
where for all i ∈ 1, k + l , F i is a non-empty forest. Here, k represents the number of nodes on the right-most branch of t and l is the number of nodes on the left-most branch of s. Definition 3.29. Let t and s be two trees which respective right comb representation and left comb representation are given above. We denote by k (respectively l) the number of nodes on the right-most (respectively left-most) branch of t (respectively s). Let σ be a (k, l)-quasi-shuffle which has for image 1, n for n a positive integer. We denote σ(t, s) the tree obtained this way:

1. We first consider the ladder with n nodes: Node 1 Node 2 Node n 2. For all i ∈ 1, k , we graft F i as the left son at the node σ(i).

3. For all i ∈ k + 1, k + l , we graft F i as right son at the node σ(i).

Example 3.30. Consider the (2, 2)-quasi-shuffle σ = (1, 3, 2, 3). Let us take

t = F 1 F 2 and s = F 3 F 4 , then σ(t, s) = F 1 F 3 F 2 F 4 .
Remark 3.31. For multiple generators, we extend the definition of this product naturally over A(X) except when a node of this ladder has left and right sons. In this case, the decoration of this node is the concatenation of the decorations of the roots of the forests grafted at this node. Remark 3.34. For all trees t, s of A and for all σ, γ ∈ QSh(k, l), we have

σ(t, s) = γ(t, s) ⇐⇒ σ = γ.
Proof . The ladder used to build σ(t, s) and γ(t, s) is a tree along the path going from the root to the nl(t)-th leaf. Comparing right and left sons of each node of the ladder we see σ = γ. ■

Description of the product

Let t, s be two trees different from |. We will suppose that our trees are written as follows:

t = F 1 F 2 • • • F k and s = F k+1 F k+2 • • • F k+l Notation 3.35. Let F = t 1 . . . t k
be a forest and t a tree. We denote by F ∨ t the following tree:

t 1 ∨ • • • ∨ t k ∨ t.
Theorem 3.36. Let t, s be two trees different from | as described above. Then

t * s = σ∈QSh(k,l) σ(t, s).
Proof . We prove this theorem by induction on the sum of k and l. As t, s ̸ = |, k, l ∈ N \ {0}. To initialize the induction, the only case to check is k + l = 2, that is to say k = 1 and l = 1. Let us remind that T 1 = and QSh(1, 1) = {(1, 1), (1, 2), (2, 1)}. Then

(1, 1) , + (2, 1) , + (1, 2) ,

= + + = + + = * .
Secondly, we check the heredity. Let (k, l) ∈ N \ {0} such that k + l ≥ 2. Let us suppose that for all couples (k ′ , l ′ ) ∈ N \ {0} such that k ′ + l ′ < k + l, we have the theorem. Then

t * s = t ≺ s + t • s + t ≻ s = F 1 ∨       F 2 F 3 • • • F k * F k+1 F k+2 • • • F k+l       +       F 1 F 2 • • • F k * F k+2 F k+3 • • • F k+l       ∨ F k+1 + F 1 ∨       F 2 F 3 • • • F k * F k+2 F k+3 • • • F k+l       ∨ F k+1 = F 1 ∨       τ ∈QSh(k-1,l) τ       F 2 F 3 • • • F k , F k+1 F k+2 • • • F k+l             + F 1 ∨       γ∈QSh(k-1,l-1) γ       F 2 F 3 • • • F k , F k+2 F k+3 • • • F k+l             ∨ F k+1 +       δ∈QSh(k,l-1) δ       F 1 F 2 • • • F k , F k+2 F k+3 • • • F k+l             ∨ F k+1 = τ ∈QSh(k-1,l) σ 1 =(1,τ (1)+1,...,τ (k+l-1)+1) σ 1 (t, s) + γ∈QSh(k-1,l-1) σ 2 =(1,γ(1)+1,...,γ(k-1)+1,1,γ(k)+1,...,γ(k+l-2)+1) σ 2 (t, s) + δ∈QSh(k,l-1) σ 3 =(δ(1)+1,...,δ(k)+1,1,δ(k+1)+1,...,δ(k+l-1)+1) σ 3 (t, s). P. Catoire Therefore, t * s = σ 1 ∈QSh(k,l) σ -1 1 ({1})={1} σ 1 (t, s) + σ 2 ∈QSh(k,l) σ -1 2 ({1})={1,k+1} σ 2 (t, s) + σ 1 ∈QSh(k,l) σ -1 3 ({1})={k+1} σ 3 (t, s) = σ∈QSh(k,l)
σ(t, s). ■

Thanks to the previous theorem, we get Corollary 3.37. Let t, s be two trees different from |. Then

t ≺ s = σ∈QSh(k,l) σ -1 (1)={1} σ(t, s), t • s = σ∈QSh(k,l) σ -1 (1)={1,k+1} σ(t, s), t ≻ s = σ∈QSh(k,l) σ -1 (1)={k+1} σ(t, s).
Proof . We just refer to the previous proof. ■ Remark 3.38. In [START_REF] Burgunder | Structure theorems for dendriform and tridendriform algebras[END_REF], the authors describe the tridendriform structure over Schröder trees with stuffle paths in Section 7.1.2. A stuffle path is a sequence σ -1 ({i}) i∈ 1,n where σ is a (k, l)quasi-shuffle and n = max(σ). As the set of quasi-shuffles is in bijection with stuffle paths, it gives an equivalent description of the tridendriform structure. But it is still defined inductively.

Remark 3.39. For many generators, we extend these definitions to decorated trees thanks to Lemma 3.31. Some quasi-shuffle algebras are studied in [START_REF] Foissy | Lie theory for quasi-shuffle bialgebras[END_REF] nearby Proposition 4. In the case of quasishuffle algebras over words, the tridendriform structure is given by shuffling letters. Otherwise, we can also define a tridendriform structure looking at from which word comes the last letter (and not the first). If we do so with Schröder trees, the relation (2.1) is not satisfied.

(3, 1)-dendriform bialgebras quotients

Definition 3.40. Let (H, ≺, •, ≻, 1, ∆, ε) be (3, 1)-dendriform bialgebra. We say that I is a (3, 1)dendriform biideal if I is a tridendriform ideal and I is a coideal of H.

With these definitions, it follows: Proposition 3.41. Let (H, ≺, •, ≻, 1, ∆, ε) be a (3, 1)-dendriform bialgebra. Let I be a (3, 1)dendriform biideal. Then H ⧸ I with the quotient structure is a (3, 1)-dendriform bialgebra.

Graded dual of A

Tridendriform coalgebra

We consider (A, * , |, ∆, ε) as a graded and connected Hopf algebra (see Definitions 3.5 and 3.6). We will now describe the graded dual of A, denoted by A ⊛ . Let us consider the bilinear map ⟨•, •⟩ : A × A → K, defined by ⟨t, s⟩ = δ t,s for t, s two trees. Through this pairing, we identify A ⊛ with A as vector spaces with Φ :

     A → A ⊛ , a → A → K, b → ⟨a, b⟩. (4.1)
Let us put for all ⋉ ∈ {≺, •, ≻, * }, ∆ ⋉ = ⋉ ⊛ . In order to be readable, we will write ∆ • instead of ∆ • . Moreover, we will write ∆ * = ∆. Then in A ⊛ + , we have

∆ = ∆ ≺ + ∆ • + ∆ ≻ .
Moreover, for all a ⊗ b ∈ A⊗A and t ∈ A ≃ A ⊛ :

⟨∆ ⋉ (t), a ⊗ b⟩ = ⟨t, a ⋉ b⟩.
By duality, choosing all the quadruples (⋉, ⋊, ⋉ ′ , ⋊ ′ ) elements of (2.8) and using the relations (2.1)-(2.7), we get in

A ⊛ + (Id ⊗ ∆) • ∆ ≺ = (∆ ≺ ⊗ Id) • ∆ ≺ , (4.2)
(Id ⊗ ∆ ≺ ) • ∆ ≻ = (∆ ≻ ⊗ Id) • ∆ ≺ , (4.3)
(Id ⊗ ∆ ≻ ) • ∆ ≻ = (∆ ⊗ Id) • ∆ ≻ , (4.4)
(Id ⊗ ∆ • ) • ∆ ≻ = (∆ ≻ ⊗ Id) • ∆ • , (4.5)
(Id ⊗ ∆ ≻ ) • ∆ • = (∆ ≺ ⊗ Id) • ∆ • , (4.6)
(Id ⊗ ∆ ≺ ) • ∆ • = (∆ • ⊗ Id) • ∆ ≺ , (4.7)
(Id ⊗ ∆ • ) • ∆ • = (∆ • ⊗ Id) • ∆ • . (4.8)
Remark 4.1. Each map described above are maps from A ⊛ + into a space where each term is well defined according to the Remark 2.6.

Example 4.2. We have

(Id ⊗ ∆) • ∆ ≺ = (Id ⊗ ∆) ⊗ | = ⊗ | ⊗ |, (∆ ≺ ⊗ Id) • ∆ ≺ = (∆ ≺ ⊗ Id) ⊗ | = ⊗ | ⊗ |.
Then we can break our coproduct ∆ over A ⊛ + in three pieces such that the relations (4.

C • K by ∆(1 C ) = 1 C ⊗ 1 C .
We also define for all x ∈ C

∆(x) = ∆(x) + 1 C ⊗ x + x ⊗ 1 C , ∆ ≺ (x) = ∆≺ (x) + x ⊗ 1 C , ∆ ≻ (x) = ∆≻ (x) + 1 C ⊗ x. As a consequence, ∆ = ∆ ≺ + ∆ • + ∆ ≻ . Note that we do not define ∆ ≺ (1 C ), neither ∆ • (1 C ), nor ∆ ≻ (1 C ).
As a consequence of these definitions, the relations (4.2)-(4.8) are satisfied over C.

(1, 3)-dendriform bialgebras

We will study the compatibilities between the product m of A ⊛ and the coproducts ∆ ≺ , ∆ • and ∆ ≻ of A ⊛ . We already know that ∆

• m = m A ⊛ ⊗A ⊛ • (∆ ⊗ ∆). Let f, g ∈ A ⊛ + and u, v ∈ A + . Then f (|) = g(|) = 0. Let ⋉ ∈ {≺, •, ≻}.
Thanks to our construction of ∆ preserving the tridendriform structure, we get

⟨∆ ⋉ • m(f ⊗ g), u ⊗ v⟩ = ⟨m(f ⊗ g), u ⋉ v⟩ = ⟨f ⊗ g, ∆(u ⋉ v)⟩ = ⟨f ⊗ g, ∆(u) ⋉ ∆(v)⟩.
For x ∈ A, we also denote ∆(x) = x ′ ⊗ x ′′ , omitting the summation symbol to shorten the writing. Therefore, the results of the Lemma 3.12 can be written as

∆(u ≺ v) = v ⊗ u + v ′ ⊗ u ≺ v ′′ + u ′ * v ⊗ u ′′ + u ′ ⊗ u ′′ ≺ v + u ′ * v ′ ⊗ u ′′ ≺ v ′′ , ∆(u ≻ v) = u ⊗ v + u ′ * v ′ ⊗ u ′′ ≻ v ′′ + u ′ ⊗ u ′′ ≻ v + u * v ′ ⊗ v ′′ + v ′ ⊗ u ≻ v ′′ , ∆(u • v) = u ′ * v ′ ⊗ u ′′ • v ′′ + u ′ ⊗ u ′′ • v + v ′ ⊗ u • v ′′ .
Notation 4.5. For all ⋉ ∈ {≺, •, ≻} and for all x ∈ A ⊛ , we denote

∆ ⋉ (x) = x ′ ⋉ ⊗ x ′′ ⋉
omitting the sum to shorten the writing. This is the generalized Sweedler's notation.

Remark 4.6. For all f ∈ A ⊛ and for all u, v ∈ A,

⟨f, u ⋉ v⟩ = f (u ⋉ v) = ∆ ⋉ (f )(u ⊗ v) = f ′ ⋉ (u)f ′′ ⋉ (v).
From those equations, we get for any f, g

∈ A ⊛ + ∆ ≺ (f g) = g ⊗ f + g ′ ≺ ⊗ f g ′′ ≺ + f ′ ≺ g ⊗ f ′′ ≺ + f ′ g ′ ≺ ⊗ f ′′ g ′′ ≺ , (4.9)
∆ • (f g) = f ′ g ′ • ⊗ f ′′ g ′′ • + f g ′ • ⊗ g ′′ • + g ′′ • ⊗ f g ′ • , (4.10) ∆ ≻ (f g) = f ⊗ g + f ′ g ′ ≻ ⊗ f ′′ g ′′ ≻ + f ′ ⊗ f ′′ g + g ′ ≻ ⊗ f g ′′ ≻ . (4.11)
We detail how to obtain (4.9). The other verifications are left to the reader,

∆ ≺ (f g), u ⊗ v = f ⊗ g, ∆(u ≺ v) = f ⊗ g, v ⊗ u + v ′ ⊗ u ≺ v ′′ + u ′ * v ⊗ u ′′ + u ′ ⊗ u ′′ ≺ v + u ′ * v ′ ⊗ u ′′ ≺ v ′′ = ⟨g ⊗ f, u ⊗ v⟩ + f ⊗ g ′ ≺ ⊗ g ′′ ≺ , v ′ ⊗ u ⊗ v ′′ + f ′ ≺ ⊗ f ′′ ≺ ⊗ g, u ′ ⊗ v ⊗ u ′′ + f ⊗ g ′ ≺ ⊗ g ′′ ≺ , u ′ ⊗ u ′′ ⊗ v + f ′ ⊗ f ′′ ⊗ g ′ ≺ ⊗ g ′′ ≺ , u ′ ⊗ v ′ ⊗ u ′′ ⊗ v ′′ = g ⊗ f + g ′ ≺ ⊗ f g ′′ ≺ + f ′ ≺ g ⊗ f ′′ ≺ + f ′ g ′ ≺ ⊗ f ′′ g ′′ ≺ , u ⊗ v .
So we get (4.9). This encourages us to establish the following definition:

Definition 4.7. A (1, 3)-dendriform bialgebra is a family (H, m, η, ∆ ≺ , ∆ • , ∆ ≻ , ε) such that
• (H, m, η) is a unitary algebra.

• (H, ∆ ≺ , ∆ • , ∆ ≻ ) is an augmented tridendriform coalgebra and ε is the counit of (H, ∆) where ∆ = ∆ ≺ + ∆ • + ∆ ≻ .

• The relations (4.9), (4.10) and (4.11) are satisfied for all f, g ∈ H.

Example 4.8. The bialgebra (A ⊛ , m, η, ∆, ε) with three maps ∆ ≺ , ∆ • and ∆ ≻ defined above is a (1, 3)-dendriform bialgebra.

Remark 4.9. If H is a graded (3, 1)-dendriform bialgebra, then its graded dual H ⊛ is a (1, 3)dendriform bialgebra. Moreover, if H is a graded (1, 3)-dendriform bialgebra, H ⊛ is a (3, 1)dendriform bialgebra.

Study of the coproduct of A ⊛

Definition 4.10 (lightning decomposition of a tree). Let t be a tree and i ∈ 1, nl(t) . We call the lightning decomposition of t on the leaf number i, the couple of trees obtained by hitting a lightning on t at the leaf number i going along the path from that leaf to the root. This lightning splits the tree into two pieces and by contracting unnecessary nodes, we obtain an ordered pair of two trees (t 1 , t 2 ).

Example 4.11.

→ → ,

This notion arises from [3, Definition 2.2 and Example 2.3], where a complete bialgebra called TSym is built with a coproduct based on the lightning decomposition of a tree. It also appears in [START_REF] Burgunder | Structure theorems for dendriform and tridendriform algebras[END_REF] where lightings are called cutting paths in Section 7.1.1. Notation 4.12. Let t be a tree and m be an element of 1, nl(t) , we denote by ( m t, t m ) the lightning decomposition of t on the leaf number m.

Let t be a tree of A. Let us consider it as an element of A ⊛ through the pairing (4.1). Let u, v be trees of A and we denote If we use a lightning on the m-th leaf of the tree t with m = nl(u), that is the ladder that we see on the drawing above, we find

u = F 1 F 2 • • • F k and v = F k+1 F k+2 • • • F k+l Then ⟨∆(t), u ⊗ v⟩ = ⟨t, u * v⟩ = σ∈QSh(k,l) ⟨t, σ(u, v)⟩ = 1 if ∃σ ∈ QSh(k, l) such that t = σ(u, v), 0
m t = u and t m = v. (4.12)
Moreover, all the elements of the form σ(u, v) for a certain σ ∈ QSh(k, l), satisfy

m σ(u, v) = u and σ(u, v) m = v,
where m = nl(u). And these trees are the only ones with this property. In fact, let us consider t a tree and m ∈ N \ {0} such that m t = u and t m = v. Then following the road of the lightning, we have the following description of t:

forests of v as a left comb, forests of u as a right comb where these forests appear "in the good order". Consequently, we deduce Proposition 4.14. The coproduct of A ⊛ is given for all tree t different from | by

∆(t) = nl(t) m=1 m t ⊗ t m .
We get combinatorial interpretations of the pieces of coproduct:

1. We interpret ∆ ≺ (t) as the sum of m t⊗t m such that the lightning starting at the leaf number m of t touches the right-most exterior branch of t.

2. The terms of ∆ ≻ (t) is the sum of these terms where the lightning touches the left-most exterior branch of t.

3. The terms of ∆ • (t) are those where the lightning does not touch any of the exterior branches. In [4, Section 7.3], we find an equivalent description of the coproducts with cutting paths.

The following results will be useful in the Section 5.2.3. Proof . Firstly, we consider four trees a, b, c, d. We denote k := nl(a) = nl(b) and l := nl(c) = nl(d). We define for any tree t with m leaves and for all n ∈ 1, m

∆ n (t) = n t ⊗ t n .
As a consequence, denoting λ k,l the number of elements of QSh(k, l) (which is not zero),

∆ k (a * b) = λ k,l a ⊗ b, ∆ k (c * d) = λ k,l c ⊗ d.
As a * b = c * d and λ k,l ̸ = 0, we deduce that a and b are collinear and c and d are collinear. Secondly, we consider a = i∈ 1,n λ i a i , b = j∈ 1,m β j b j , c = q∈ 1,r γ q c q and d = l∈ 1,s δ l d l where the λ i , β j , γ q , δ l are all non zero scalars and a i , b j , c q , d l are trees. Expanding the writings of a * b and c * d and applying ∆ k , we also obtain

∆ k (a * b) = λ k,l a ⊗ b, ∆ k (c * d) = λ k,l c ⊗ d.
We also have the same conclusion as in the first case. ■

Following the same ideas, we also have Theorem 4.18. Let X ∈ A k ⊗ A l for some pair of non-negative integers (k, l). Let X = i∈I a i ⊗ b i be a minimal writing of X for the number of terms. If there exists ⋉ ∈ {≺, •, ≻, * } such that i∈I a i ⋉ b i = 0, then X = 0.

Proof . Let ⋉ ∈ {≺, •, ≻, * } such that i∈I a i ⋉ b i = 0. We define ∆ k as in the previous proof. As a consequence,

0 = ∆ k (X) = ∆ k i∈I a i ⋉ b i = λ ⋉ (k,l) i∈I a i ⊗ b i , where λ ≺ (k,l) , λ • (k,l) , λ ≻ (k,l)
are respectively the number of elements of QSh(k, l) such that σ -1 ({1}) = {1}, σ -1 ({1}) = {1, k + 1} and σ -1 ({1}) = {k + 1}. None of these numbers are equal to zero. But this sum is 0. As a consequence, at least one family (a i ) i∈I or (b i ) i∈I is linearly dependant. However X = i∈I a i ⊗ b i is a minimal writing. So X = 0. ■

Description of the product of A ⊛

Let us remind the definitions of ¡ and m TSym from [3, Definitions 2.9 and 4.1].

Definition 4.19. Let t be a tree with k leaves and consider k trees t 1 , . . . , t k . We denote by t ¡ (t 1 , . . . , t k ) the tree obtained by grafting t i on the i-th leaf of t for each i ∈ 1, k . We can also use multiple lightnings on a single tree t. If l is a non-negative integer, we denote by Light l (t) ⊆ A l+1 the set of all the decompositions of t obtained using l successive lightnings. Consider another tree s such that deg(s) = r. We define m TSym (s ⊗ t) := (t 1 ,...,t r+1 )∈Light r (t)

s ¡ (t 1 , . . . , t r+1 ).

Examples 4.20.

Here is an example of multiple lightnings:

→ , → , , , Light 2 = |, |, , |, , | , , |, | , |, , , |, , , , , | , , , | , , |, , , |, , , , .
Here is an example for the product:

¡ , = , m TSym , = + + + + + + + + + .
Notation 4.21. Consider a vector space V and a linear map m :

V ⊗ V → V , we denote m op the linear map from V ⊗ V to V such that for all a ⊗ b ∈ V ⊗ V m op (a ⊗ b) = m(b ⊗ a).
If (A, m, 1) is an algebra, we denote A op the algebra (A, m op , 1).

Proposition 4.22. The product m of A ⊛ is the product of m op TSym .

Proof . Let (u, v) ∈ T n × T m ⊂ A ⊛ × A ⊛ for m and n some non-negative integers. Let w be a tree of A. In this case, thanks to Theorem 3.20

⟨m(u ⊗ v), w⟩ = ⟨u ⊗ v, ∆(w)⟩ = c admissible cut of w δ u⊗v (G c (w) ⊗ R c (w)).
Therefore, this quantity is not equal to zero if and only if w = v ¡ (w 1 , . . . , w deg(v)+1 ) and u

occurs in G c (w) with c the cut of w such that R c (w) = v. But G c (w) = w 1 * • • • * w deg(v)+1
, by Theorem 3.36 and Remark 4.12, we get that G c (w) is the sum of all the trees which gives (w 1 , . . . , w deg(v)+1 ) after deg(v) well-placed lightnings. As a consequence, we have

m(u ⊗ v) = u →(u 1 ,...,u deg(v)+1 ) v ¡ (u 1 , . . . , u deg(v)+1 ) = m TSym (v ⊗ u) = m op TSym (u ⊗ v). ■
Thanks to the previous proposition, Proposition 4.14 and Remark 4.15, we deduce that Theorem 4.23. The bialgebras A ⊛ and TSym op are the same.

Remark 4.24. We can give TSym a (1, 3)-dendriform bialgebra structure given in Proposition 4.14.

Remark 4.25. With multiple generators, we have A(D) ⊛ = TSym(D) op . In the bialgebra TSym(D) its coproduct is an expansion of the lightning decomposition over elements of n≥0 T g n (T (D)). Let t ∈ T g n (T (D)). We consider a lightning on one of its leaves. If a node of t only appears on the left tree or only on the right one, then its decoration stays unchanged. Otherwise denoting d 1 . . . d k its decoration and l the edge number from which the lightning arrives, the decoration of this node on the left tree is d 1 . . . d l-1 and on the right tree it is d l . . . d k .

Example 4.26. We consider the following tree:

(3, 2)-dendriform bialgebras 5.1 Definition

Let (A, ≺, ≻, •) be a tridendriform algebra. The definitions of dendriform algebras, dendriform coalgebras, dendriform bialgebras, codendriform bialgebras and bidendriform bialgebras (that we will call (2, 2)-dendriform bialgebras) can be found in the article [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF].

Definition 5.1. A (3, 2)-dendriform bialgebra is a sextuple (A, ≺, •, ≻, ∆ ← , ∆ → ) such that • (A, ≺, •, ≻) is an augmented tridendriform algebra. • (A, ∆ ← , ∆ → ) is an augmented dendriform coalgebra.
• The following relations are satisfied for all a, b ∈ A:

∆ ← (a • b) = a ′ b ′ ← ⊗ a ′′ • b ′′ ← + b ′ ← ⊗ a • b ′′ ← , (5.1)
∆ → (a • b) = a ′ b ′ → ⊗ a ′′ • b ′′ → + a ′ ⊗ a ′′ • b + b ′ → ⊗ a • b ′′ → , (5.2) ∆ ← (a ≺ b) = a ′ b ′ ← ⊗ a ′′ ≺ b ′′ ← + a ′ b ⊗ a ′′ + b ′ ← ⊗ a ≺ b ′′ ← + b ⊗ a, (5.3)
∆ → (a ≺ b) = a ′ b ′ → ⊗ a ′′ ≺ b ′′ → + a ′ ⊗ a ′′ ≺ b + b ′ → ⊗ a ≺ b ′′ → , (5.4)
∆ → (a ≻ b) = a ′ b ′ → ⊗ a ′′ ≻ b ′′ → + a ′ ⊗ a ′′ ≻ b + b ′ → ⊗ a ≻ b ′′ → + ab ′ → ⊗ b ′′ → + a ⊗ b, (5.5)
∆ ← (a ≻ b) = a ′ b ′ ← ⊗ a ′′ ≻ b ′′ ← + a ′ ⊗ a ′′ ≻ b + ab ′ ← ⊗ b ′′ ← .
(5.6)

Remark 5.2. The definition of (3, 2)-dendriform bialgebra is made such that (A, ≺, •, ≻,

∆ ← , ∆ → ) is a (3, 2)-dendriform bialgebra ⇐⇒ (A, ≺, •+ ≻, ∆ ← , ∆ → ) and (A, ≺ +•, ≻, ∆ ← , ∆ → ) are (2, 2)-dendriform bialgebras.
To check that this definition is consistent, we will give an example of such an object.

5.2 Example of a (3, 2)-dendriform bialgebra

Description

To find an example, we consider the (3, 1)-dendriform bialgebra given by A. Our objective is to break the coproduct ∆ into two pieces, denoted by ∆ → and ∆ ← , which will satisfy the desired relations. Let us consider t a tree and let us decorate the right-most leaf by a symbol d.

Example 5.3. For t = , this gives d .

Definition 5.4. Let t be a tree of A different from |. We define ∆ ← (t) as the sum of the terms from ∆(t) which gets the decorated leaf by d (that means the right-most leaf of t) on the left tensor, i.e., on G c (t). Then we delete the decoration. We define ∆ → (t) as the sum of the terms from ∆(t) which gets the decorated leaf by d on the right tensor, i.e., on R c (t). Then we delete the decoration. So, ∆(t) = ∆ ← (t) + ∆ → (t). Notation 5.5. Let t be a tree. We denote t d the tree t which last leaf is decorated with d. As we do not talk about lightnings in this section, we use again this notation during this section. Proposition 5.6. We consider (A, ≺, •, ≻) with its tridendriform algebra structure, we build the following coproducts:

∆← (t) = c admissible cut of t the right-most leaf of t is cut G c (t) ⊗ R c (t) -t ⊗ 1, ∆→ (t) = c admissible cut of t the right-most leaf of t is not cut G c (t) ⊗ R c (t) -1 ⊗ t.
Then (A, ≺ +•, ≻, ∆ ← , ∆ → ) and (A, ≺, •+ ≻, ∆ ← , ∆ → ) are bidendriform bialgebras. As a consequence, (A, ≺, •, ≻, ∆ ← , ∆ → ) is a (3, 2)-dendriform bialgebra.

Proof . We will now show that (A, ≺, •, ≻, ∆ ← , ∆ → ) satisfies the Definition 5.1. The first constraint is satisfied. For the second one, we need to check the three following equalities:

∆← ⊗ Id • ∆← = Id ⊗ ∆ • ∆← , (5.7) ∆→ ⊗ Id • ∆← = Id ⊗ ∆← • ∆→ , (5.8) ∆ ⊗ Id • ∆→ = Id ⊗ ∆→ • ∆→ .
(5.9)

We prove each case by following the destination of the decorated leaf with d and using that ∆ = ∆← + ∆→ to obtain those equalities. More precisely, let us consider a tree t of A. In order to show (5.7), we look for all the terms of ∆ ⊗ Id • ∆ t d and Id ⊗ ∆ • ∆ t d where the decorated leaf is on the left tensor. In the first term, we find ∆← ⊗ Id • ∆← t d , then in the second, we find Id ⊗ ∆ • ∆← t d . Finally, the coassociativity property gives that ∆← ⊗ Id • ∆← t d = Id ⊗ ∆ • ∆← t d . We proceed the same way with (5.8) and (5.9). For the last condition, we remind that for all a, b ∈ A and for all ⋉ ∈ {≺, •, ≻}

∆(a ⋉ b) = ∆(a) ⋉ ∆(b) = (∆ → (a) + ∆ ← (a)) ⋉ (∆ → (b) + ∆ ← (b)) = ∆ ← (a) ⋉ ∆ ← (b) + ∆ ← (a) ⋉ ∆ → (b) + ∆ → (a) ⋉ ∆ ← (b) + ∆ → (a) ⋉ ∆ → (b).
As a consequence, identifying the terms which get the last leaf of a ⋉ b at the left or at the right of the tensor, we obtain for all ⋉ ∈ {≺, •, ≻}

∆ → (a ⋉ b) = ∆ ← (a) ⋉ ∆ → (b) + ∆ → (a) ⋉ ∆ → (b) = ∆(a) ⋉ ∆ → (b), ∆ ← (a ⋉ b) = ∆ ← (a) ⋉ ∆ ← (b) + ∆ → (a) ⋉ ∆ ← (b) = ∆(a) ⋉ ∆ ← (b).
We need to check the six relations in the definition of (3, 2)-dendriform bialgebra. I will only detail the computation for (5.4), the other verifications are left to the reader. Let a, b ∈ A, then

∆→ (a ≺ b) = (1 ⊗ a + a ⊗ 1 + a ′ ⊗ a ′′ ) ≺ (1 ⊗ b + b ′ → ⊗ b ′′ → ) -1 ⊗ a ≺ b = 1 ⊗ a ≺ b + b ′ → ⊗ a ≺ b ′′ → + a ′ ⊗ a ′′ ≺ b + a ′ * b ′ → ⊗ a ≺ b ′′ → -1 ⊗ a ≺ b = a ′ * b ′ → ⊗ a ′′ ≺ b ′′ → + a ′ ⊗ a ′′ ≺ b + b ′ → ⊗ a ≺ b ′′ → . ■

Counting codendriform primitives

In order to study the primitives of the (3, 2)-dendriform bialgebra A, we will use the results from the article [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF]. Let us remind results from [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF] without proofs. Let (C, ∆ ← , ∆ → ) be a dendriform coalgebra. We introduce the following sets defined by induction:

P C (0) = {Id C }, P C (1) = {∆ ← , ∆ → } ⊆ L(C, C ⊗2 ), P C (n) = Id ⊗(i-1) ⊗ ∆ ← ⊗ Id ⊗(n-i) • P | P ∈ P C (n -1), i ∈ 1, n ∪ Id ⊗(i-1) ⊗ ∆ → ⊗ Id ⊗(n-i) • P | P ∈ P C (n -1), i ∈ 1, n ⊆ L C, C ⊗(n+1) .
Definition 5.9. We say that a dendriform coalgebra C is connected if for all a ∈ C, there exists n a ∈ N such that for all P ∈ P C (n a ), P (a) = 0.

With these definitions, we get Lemma 5.10. The coalgebra (A, ∆ ← , ∆ → ) is connected.

Proof . We notice that for all ▷ ∈ {←, →}, we have for all n ∈ N ∆

▷ (A n ) ⊆ n-1 k=1 A n-k ⊗ A k .
Therefore, choosing t ∈ A n , p ∈ N and δ ∈ P C (p), we get

δ(A n ) ⊆ k 1 +•••+k p+1 =n k 1 ,...,kp+1>0 A k 1 ⊗ • • • ⊗ A k p+1 .
So, for all p ≥ n and for all δ ∈ P C (p), δ(A n ) = (0). ■

Following [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF], we get Theorem 5.11. Let H be a (2, 2)-dendriform bialgebra, which we assume to be connected as a dendriform coalgebra. Then H is generated by Prim Codend (H) as a dendriform algebra.

Proof . See [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF]Theorem 21]. ■

This theorem also applies to (A, ≺ +•, ≻, ∆ ← , ∆ → ). So, A is generated by Prim Codend (A) with the operations ≺ +•, ≻. In particular, A is generated by Prim Codend (A) as a tridendriform algebra. More generally, this gives Proposition 5.12. Let H be a connected (as a dendriform coalgebra) (3, 2)-dendriform bialgebra. Then H is generated by Prim Codend (H) as a tridendriform algebra. Remark 5.13. By [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF], we have

A = Prim Codend (A) ⊕ (A ⪯ A + A ≻ A) = Prim Codend (A) ⊕ (A ≺ A + A ⪰ A) = Prim Codend (A) + (A ≺ A + A • A + A ≻ A).
Theorem 5.14. Let H be a graded (2, 2)-dendriform bialgebra by N such that for all n ∈ N, H n is finite dimensional and H 0 = (0). We consider the formal series

P (X) = +∞ n=1 dim(Prim Codend (H) n )X n , R(X) = +∞ n=1 dim(H n )X n . Then P (X) = R(X) (1 + R(X)) 2 .
Proof . See [START_REF] Foissy | Bidendriform bialgebras, trees, and free quasi-symmetric functions[END_REF]Corollary 37]. ■ Remark 5.15. We may ask ourselves why we want to compute codendriform primitives. We already have the generator . But it is a generator for the tridendriform structure meanwhile Prim Codend (A) generates A as a dendriform algebra for both (≺ +•, ≻) and (≺, •+ ≻).

All the theorems cited above are valid for (A + , ≺, •, ≻, ∆ ← , ∆ → ). Let us compute the formal series:

R(X) = +∞ n=1 dim(A n )X n .
P. Catoire Definition 5.16 (non-redundant bracketing). Let n ∈ N. Let w be a word over the alphabet {|} of n + 1 symbols. We call non-redundant bracketing of w all insertions of pairs of parenthesis in w which do not contain subwords of the form (|) and such that w is not between brackets. We will denote the set of non-redundant bracketing of any word w by P n . The set of non-redundant bracketings of all words is denoted by P.

Remark 5.17. We have P = n≥0 P n . Moreover, we will denote T := n≥0 T n .

It is known that dim(

A n ) = |P n |.
Referring to [START_REF] Sloane | The on-line encyclopedia of integers sequences[END_REF], we know that |P n | = a n where a n is the n-th second Schröder's problem number. The numbers a n are called small Schröder's numbers or super Catalan numbers. With this reference, we also have

∀n ∈ N, n ≥ 2, (n + 1)a n = (6n -3)a n-1 -(n -2)a n-2 and a 0 = 1, a 1 = 1.
Thanks to this, with the help of a computer we get R(X) = X + 3X Other properties of small Schröder's numbers are given in [START_REF] Sloane | The on-line encyclopedia of integers sequences[END_REF].

Looking at the first terms of P , we suspect that P is the series of large Schröder's numbers denoted by (A n ) n∈N defined by A 0 = 0, A 1 = 1 = A 2 , then for all n ≥ 3, A n = 2a n-2 .

Remark 5.19. In particular, we want to show

P (X) = X + X 2 + 2X 2 R(X).
Looking once more at [START_REF] Sloane | The on-line encyclopedia of integers sequences[END_REF], we find that the formal series of R(X) is

1 + R(X) = 1 + X - √ 1 -6X + X 2 4X .
Referring to [START_REF] Sloane | The on-line encyclopedia of integers sequences[END_REF] and using Theorem 5.14, we obtain Proposition 5.20.

P (X) = +∞ n=0 A n X n where A 0 = 0, A 1 = A 2 = 1 and for all n ≥ 3, A n = 2a n-2 .
By construction of A 1 × B 1 , hypotheses of Theorem 4.18 are satisfied. As a consequence, (a ′ * b ′ → ) (a ′ ,b ′ → )∈A 1 ×B 1 is a free family. Following the same method, we have that the family (a ′′ ≺ b ′′ → ) (a ′′ ,b ′′ → )∈A 2 ×B 2 is also free. We can complete each of these families into respective basis of T deg(B 1 )+deg(A 1 ) and T deg(B 2 )+deg(A 2 ) . Using elements f , g of the dual basis defined by f (a ′ * b ′ ← ) = 1 for one and only one element of (a

′ * b ′ ← ) a ′ ∈A 1 ,b ′ ← ∈B 1 and g(a ′′ •b ′′ ← ) a ′′ ∈A 2 ,b ′′ ← ∈B 2 = 1
for one and only one element of (a ′′ • b ′′ ← ). As a consequence, with a good choice for (f, g), we get

0 = (f ⊗ g) A B a ′ * b ′ ← ⊗ a ′′ • b ′′ ← = 1.
This is a contradiction. We then deduce that ∆(a) = 0 or ∆ ← (b) = 0.

For the first equivalence, we use similar ideas. ■

This allows us to say

a • b ∈ Prim Codend (A) ⇐⇒ (b ∈ Prim Codend (A) and a ∈ Prim Coass (A)).
This defines a map

θ • : Prim Coass (A) ⊗ Prim Codend (A) → Prim Codend (A), a ⊗ b → a • b.
Using similar arguments, we also prove Proposition 5.23. For all a, b ∈ A,

∆ ← (a ≻ b) = 0 ⇐⇒ (a ∈ Prim Coass (A) and b ∈ Prim ← (A)), ∆ → (a ≺ b) = 0 ⇐⇒ (a ∈ Prim Coass (A) and b ∈ Prim → (A)).
The proposition above gives two maps

θ ≻ : Prim Coass (A) ⊗ Prim ← (A) → Prim ← (A), a ⊗ b → a ≻ b, θ ≺ : Prim Coass (A) ⊗ Prim → (A) → Prim → (A), a ⊗ b → a ≺ b.
Moreover, the space Prim Codend (A) is an algebra with •.

Definition 5.24. A dipterous algebra (H, * ) is an associative algebra with an additional bilinear operation denoted by ≺ such that, for all a, b, c ∈ H,

(a ≺ b) ≺ c = a ≺ (b * c).
We notice that A is also a dipterous algebra, as a consequence A is cofree referring to [START_REF] Loday | Algèbres de Hopf colibres[END_REF]. In other words, there exists a vector space V such that coT(V ) ≃ A as coalgebras. For more details see [START_REF] Loday | Algèbres de Hopf colibres[END_REF].

Series of coassociative primitives: using the results of the previous paragraph and the results 59 (i.e., Prim(coT(V )) = V ) and 46 (i.e, F T (V ) (X) = 1 1-F V (X) where F B is the formal series of the graded space B such that B 0 = (0)) from [START_REF] Foissy | Algèbres de Hopf combinatoires[END_REF], we obtain

F T (V ) (X) = F A (X) = 1 1 -F Prim Coass (A) (X) , which implies F Prim(A) (X) = F A (X)-1 F A . But, F A (X) = 1 + R(X).
As a consequence, we find

F Prim Coass (A) (X) = R(X) 1 + R(X)
.

P. Catoire

Proposition 5.25. We have

F Prim Coass (A) (X) = P (X) X -1.
In other words,

F Prim Coass (A) (X) = +∞ n=1 A n+1 X n ,
where A 0 = 0, A 1 = 1, A 2 = 1 and for all n ≥ 3, A n = 2a n-2 .

Proof . From (5.10), we get

F Prim Coass (A) (X) = R(X) 1 + R(X) = X + 2XR(X).
And we also have, according to Remark 5.19,

P (X) X -1 = X + X 2 + 2X 2 R(X) X -1 = X + 2XR(X). ■
Finally, we obtain Theorem 5.26. For all n ∈ N, we define

θ n : Prim Coass (A) n ⊗ → Prim Codend (A) n+1 , a ⊗ → a • .
Then for all n ∈ N, θ n is an isomorphism of vector spaces. Consider (a i ) i∈ 1,A n+1 a basis of Prim Coass (A) n . According to Theorem 4.18, the following family a i • i∈ 1,A n+1 is free in Prim Codend (A) n+1 . For reasons of dimensions, we conclude that θ n is an isomorphism. ■ Remark 5.27. We fix (A, ≺, ⪰, ∆ → , ∆ ← ) a (2, 2)-dendriform bialgebra structure. For all vector spaces V over K, we denote by Dend(V ) the free dendriform algebra generated by V . Referring to [START_REF] Ronco | Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras[END_REF]Theorem 4.6], the brace algebras B(V ) and Prim Coass (Dend(V )) are isomorphic for any vector space V . For more information about brace algebra, we refer to [START_REF] Chapoton | Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces[END_REF][START_REF] Foissy | Free brace algebras are free pre-Lie algebras[END_REF]. Thanks to [9, Theorem 35 and Corollary 36], we have A = Tridend = Dend(Prim Codend (A)). As a consequence, we can compute recursively Prim Coass (A) and Prim Codend (A) the following way:

1. Suppose we know Prim Coass (A) n . Case 2 : k = 1. As a consequence, t = t (0) ∨ t (1) where each of these trees owns at least one leaf. But, as t is not binary either t (0) is not binary, either t (1) is not binary. In the two cases, each of these trees has strictly less leaves than t. By the induction hypothesis, we can suppose that t (0) ∈ A + • A + without loss of generality. But, A + • A + is a tridendriform ideal. This implies that t ∈ A + • A + because t = t (0) ≻ ≺ t (1) . ■

Compute Prim

This allows us to conclude that A ⧸ ⟨A + • A + ⟩ and n≥0 K PBT n are isomorphic as vector spaces. Finally, the quotient space of A by A + • A + is a (3, 2)-dendriform bialgebra with its natural structure of quotient space. But, the product • is 0 on this space. In particular, the bialgebra we just obtained can be seen as a (2, 2)-dendriform bialgebra. More precisely, in A ⧸ ⟨A + • A + ⟩ * =≺ + ≻ . This product corresponds to the one written in [START_REF] Loday | Hopf algebra of the planar binary trees[END_REF]Proposition 3.2]. Moreover, it is not hard to see that the coproduct of the quotient space is the same as the one in [START_REF] Bergeron | Hopf algebras of parking functions and decorated planar trees[END_REF]Proposition 3.3]. This algebra is the Loday-Ronco algebra. Referring to [START_REF] Loday | Hopf algebra of the planar binary trees[END_REF], the Proposition 3.2 says that for t, t ′ two binary trees written t = t 1 ∨ t 2 and t ′ = t ′ 1 ∨ t ′ 2 , we have

t * t ′ = t 1 ∨ (t ′ 1 * t 2 ) + (t * t ′ 1 ) ∨ t ′ 2 .
This is exactly the descriptions of the products ≺ and ≻ restricted to binary trees.

Proposition 5.30. The coproduct given in [START_REF] Loday | Hopf algebra of the planar binary trees[END_REF] on a tree t = t 1 ∨ t 2 with t 1 ∈ T n and t 2 ∈ T m is defined by

∆(t) = j,k t (1)(j) 1 * t (1)(k) 2
⊗ t

(2)(n-j) 1

∨ t

(2)(m-k) 2

+ t ⊗ |,
where ∆(t 1 ) = n j=0 t

(1)(j) 1

⊗ t

(2)(n-j) 1

and ∆(t 2 ) = m k=0 t

(1)(k) 2

⊗ t

(2)(m-k) 2

. This coproduct is the same as the one on A ⧸ ⟨A + • A + ⟩ .

Proof . We will proceed by induction over n the number of leaves of t to show the coproduct ∆ of Loday-Ronco algebra is the same as the one over A ⧸ ⟨A + • A + ⟩ denoted by ∆. Heredity: let n ∈ N such that n ≥ 3, suppose that for all n ′ < n we have the equality between ∆ and ∆ for all trees with n ′ leaves. We check that it is also true for trees with n leaves. Let us write t = t 1 ∨ t 2 such that t 1 has n 1 leaves and t 2 has n 2 leaves. Two mutually excluding cases can happen:

• The case n 2 + 1 ̸ = n. Using the properties of ∆, we get

∆(t) = ∆(t 1 ∨ t 2 ) = ∆(t 1 ≻ (| ∨ t 2 )) = ∆(t 1 ) ≻ ∆(| ∨ t 2 ).
Applying the induction hypothesis once, we get

∆(t) = ∆(t 1 ) ≻ ∆(| ∨ t 2 ) = n 1 j=0 t (1)(j) 1 ⊗ t (2)(n 1 -j) 1 ≻ | ∨ t 2 ⊗ | + c admissible cut of t 2 G c (t 2 ) ⊗ | ∨ P c (t 2 ) = n 1 j=0 t (1)(j) 1 ⊗ t (2)(n 1 -j) 1 ≻ | ∨ t 2 ⊗ | + n 2 k=0 t (1)(k) 2 ⊗ | ∨ t (2)(n 2 -k) 2 = j,k t (1)(j) 1 * t (1)(k) 2 ⊗ t (2)(n 1 -j) 1 ∨ t (2)(n 2 -k) 2 + t ⊗ |.
So, in this case we have equality between ∆ and ∆.

• The case n 1 + 1 ̸ = n is done the same way. That is why we omit some details,

∆(t) = ∆((t 1 ∨ |) ≺ t 2 ) = j,k t (1)(j) 1 * t (1)(k) 2 ⊗ t (2)(n 1 -j) 1 ∨ | ≺ t (2)(n 2 -k) 2 + k (t 1 ∨ |) * t (1)(k) 2 ⊗ | ≺ t (2)(n-k) 2 = j,k t (1)(j) 1 * t (2)(k) 2 ⊗ t (2)(n 1 -j) 1 ∨ t (n 2 -k) 2 + t ⊗ |.
Therefore, we have the result. ■

Conclusion

Thanks to [START_REF] Loday | Trialgebras and families of polytopes[END_REF], we have given a bialgebra structure to the free tridendriform algebra A such that it is a (3, 2)-dendriform bialgebra structure. Considering the graded dual of A, we have defined (2, 3)-dendriform bialgebras. Using [START_REF] Bergeron | Hopf algebras of parking functions and decorated planar trees[END_REF], we have identified the graded dual of A with TSym as bialgebras. Moreover, it shows that TSym can be given a (2, 3)-dendriform bialgebra structure. It shows the interactions between the combinatorics on trees and the dendriform and tridendriform algebras. Some ideas are not explored in this paper. For example, can we get other quotient spaces from the free (3, 2)-dendriform bialgebra? Can we find a description of the coassociative primitives of this algebra obtained with Remark 5.27? Or can we generalize our ideas to other (n, m)dendriform structures and maybe give a more general definition?

Example 3 .

 3 30. Consider σ = (1, 3, 2, 3) a (2, 2)-quasi-shuffle.Let us take t = F 1

F 2 and s = F 3

 3

F 4 . 1 F 3 F 2 F 4 .

 41324 Then σ(t, s) = F Theorem 3.36. Let t, s be two trees different from |. Then t * s = σ∈QSh(k,l)

  Given A, B and C three augmented tridendriform algebras, do (A ⊗ B) ⊗ C and A ⊗ (B ⊗ C) have the same tridendriform algebra structure? From the first section, (A ⊗ B) ⊗ C and A ⊗ (B ⊗ C) have the same tridendriform algebra structure. There is still to check what happens with units. P. Catoire Lemma 2.12. For A, B and C three augmented tridendriform algebras, we have A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C as tridendriform algebras.

Example 3 .

 3 32. Let a, b ∈ D, then (1, 1) a , b = ab . Notation 3.33. Given F = t 1 . . . t n a forest, we define nl(

Definition 4 . 4 .

 44 2)-(4.8) are verified. In the following, we define as in [4, Definition 5.3]: Definition 4.3. A tridendriform coalgebra is a coalgebra C, ∆ , where there exist three maps ∆≺ : C → C ⊗ C, ∆ • : C → C ⊗ C, ∆≻ : C → C ⊗ C, such that the relations (4.2)-(4.8) are true for all (x, y, z) ∈ C 3 replacing each ∆ ⋉ by ∆⋉ for ⋉ ∈ {≺, •, ≻, * }. We also have ∆ = ∆≺ + ∆ • + ∆≻ . Let C, ∆≺ , ∆ • , ∆≻ be a tridendriform coalgebra. The associated augmented tridendriform coalgebra is the vector space C = 1 C • K ⊕ C where we expand our coproduct ∆ as a coproduct ∆ on 1

Remark 4 . 15 .

 415 It is the coproduct from TSym = n≥0 KP T n described in [3, Definition 4.1].

Corollary 4 . 16 .

 416 Let a, b, c, d be four homogeneous elements of A. We put deg(a) = deg(c) and deg(b) = deg(d). We suppose that a * b = c * d for the product of A. Then a and b are collinear and c and d are collinear.

Corollary 4 . 17 .

 417 Let a, b, c, d be four homogeneous elements of A. We put deg(a) = deg(c), deg(b) = deg(d). We also suppose a ≺ b = c ≺ d or a • b = c • d or a ≻ b = c ≻ d. Then a and b respectively c and d are collinear.

Definition 5 . 7 .Remark 5 . 8 .

 5758 Let (C, ∆ ← , ∆ → ) be a dendriform coalgebra. We definePrim ← (C) := c ∈ C | ∆← (c) = 0 , Prim → (C) := c ∈ C | ∆→ (c) = 0 , Prim Coass (C) := c ∈ C | ∆(c) = 0 , Prim Codend (C) := c ∈ C | ∆← (c) = ∆→ (c) = 0 . In particular, Prim Codend (C) = Prim ← (C) ∩ Prim → (C) ⊆ Prim Coass (C).

Example 5 . 18 .

 518 Let us take the word w = |||, the non-redundant bracketings of w are |(||), (||)| and |||. So |P 2 | = 3.

Proof.

  Let n ∈ N. By Proposition 5.25, we know that dim (PrimCoass (A) n ) = A n+1 . From Proposition 5.20, we obtain dim (Prim Codend (A) n+1 ) = A n+1 . But dim Prim Coass (A) n ⊗ = dim(Prim Codend (A) n+1 ).

  Codend (A) n+1 with Theorem 5.26.

3 .

 3 Use B(Prim Codend (A)) in order to compute Prim Coass (A) n+1 from Prim Coass (A) k , k ≤ n and Prim Codend (A) n+1 .

  Initialization: if n = 1, i.e., t = |, then ∆(|) = ∆(|) = | ⊗ |. If n = 2, i.e., t = , we get ∆ = ⊗ | + | ⊗ , ∆ = | * | ⊗ | ∨ | + ⊗ | = ⊗ | + | ⊗ .

  Remark 4.[START_REF] Loday | On the algebra of quasi-shuffles[END_REF]. By Remark 3.34, if such a quasi-shuffle exists, it is necessarily unique.As a consequence, if this quantity is non-zero, this implies that t has the following form:

	the forests F 1 ,...,F k are ordered,	the forests F k+1 ,...,F k+l are ordered,
	from below to top	from below to top

else.

  2 + 11X 3 + 45X 4 + 197X 5 + 903X 6 + 4279X 7 + 20793X 8 + 103049X 9 + 518859X 10 + 2646723X 11 + 13648869X 12 + 71039373X 13 + 372693519X 14 + 1968801519X 15 + 10463578353X 16 + 55909013009X 17+ 300159426963X 18 + 1618362158587X 19 + 8759309660445X 20 + • • • , P (X) = X + X 2 + 2X 3 + 6X 4 + 22X 5 + 90X 6 + 394X 7 + 1806X 8 + 8558X9 + 41586X 10 + 206098X 11 + 1037718X 12 + 5293446X 13 + 27297738X 14 + 142078746X 15 + 745387038X 16 + 3937603038X 17 + 20927156706X 18

+ 111818026018X 19 + 600318853926X

  20

Acknowledgements

I want to thank all members of the LMPA at Université du Littoral Côte d'Opale for welcoming me to prepare my PhD. I especially thank Loïc Foissy, my PhD advisor. I thank the other PhD students of the LMPA for useful discussions which give me some ideas. I also want to thank the referees for the careful reading of this paper. The author acknowledges support from the grant ANR-20-CE40-0007 Combinatoire Algébrique, Résurgence, Probabilités Libres et Opérades.

Proof . To prove this proposition, we provide an intermediate computation

Now we compute the first term of the right part of this equality:

Finally,

As a consequence, we deduce the two following equalities:

Codendriform primitives generation

Now we exactly know the dimensions of codendriform primitives. For degree 1 and 2, we have

Remark 5.21. Note that -is an element of Prim Coass (A). However,

So, it is not an element of Prim Codend (A).

From equations (5.1)-(5.6) we deduce the following equivalences: P. Catoire Proposition 5.22. For all a, b ∈ A, we have

Proof . The =⇒ implications come straightforward from the codendriform relations. For the other implication of the second equivalence, let us consider a, b ∈ A such that ∆ ← (a • b) = 0. Then we suppose by contradiction that ∆ ← (b) ̸ = 0. Without loss of generality, we can suppose that the writings of ∆ ← (b) as a sum of tensors are minimal for the number of terms.

Case 1 : ∆(a) = 0. Using codendriform relations, we know that

Let also denote by B 1 and B 2 the respective projections of B over its first and second component.

← is minimal for the degree. For reasons of degrees, we must have

This is a contradiction. As a consequence, we find ∆ ← (b) = 0. We define B the same way as the previous case. We take back the notations for projections over first and second component of B and we expand them for A. The tensors of ∆ ← (a • b) from which the first term has maximal degree are

Without loss of generality, we can suppose that the writings of ∆ ← (b) and ∆(a) are minimal for the number of terms. We check (a

are free families of A. For this, we will use the Theorem 4.18. In fact, we consider for all (a

Quotient of A by A + • A +

Consider A the free tridendriform algebra with one generator. We remind that we write A = 1 A K ⊕ A + . We define A + • A + as the tridendriform ideal generated by A + • A + .

Lemma 5.28. The tridendriform ideal

Proof . By definition, it is a tridendriform ideal.

We check I := A + • A + is a dendriform coideal. First, as ε(|) = 1 and is zero on A + , it is easily seen that ε(I) = (0). Then for all s 1 , s

As a consequence, ∆(s)

In both expressions, each term is either 0 or an element of A ⊗ I + I ⊗ A using the fact that I is a tridendriform ideal. We have proved that ∆(x⋉y) and ∆(y ⋊x) belong to A⊗I +I ⊗A. Using similar ideas, we easily prove the same results for ∆

So the proof is complete. ■

As a consequence, we can divide A by the (3, 2)-dendriform biideal A + • A + . Referring to the Remark 3.10, we will give a characterization to describe the trees in A + • A + . Lemma 5.29. Let t be a tree of A. Then t is not a binary tree if and only if

So, this is clear that t is not a binary tree.

Conversely, consider a non-binary tree t. We want to show that t has at least one writing with •. To prove this, we proceed by induction on the number of leaves of t. We denote n the number of leaves of t. To initialize the induction, we need to begin with n = 3. The only tree which is not binary is = • . To prove the heredity, we consider that n > 3 and we suppose that for all 3 ≤ m < n, non-binary trees with m leaves are elements of A + • A + . Let t be a tree with n leaves. We have the following writing:

Case 1 : k ≥ 2. We choose i ∈ 1, k -1 . Then writing t 1 = t (0) ∨ • • • ∨ t (i) ∨ | and t 2 = t (i+1) ∨ • • • ∨ t (k) , we get


Sébastien Lalanne
email: sebastien.lalanne@chu-rennes.fr

Vincent Cattoir

Francois Guerin

Marie-Clémence Verdier

Matthieu Revest

François Guerin

Differential response to antibiotic therapy in staphylococcal infective endocarditis: contribution of an ex vivo model

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Staphylococcus aureus is the leading cause of Infective endocarditis (IE), and accounts for 40% of cases of IE [START_REF] Wang | Management Considerations in Infective Endocarditis: A Review[END_REF] . S. aureus IE is characterized by the occurrence of early heart valve damage, and is associated with a high mortality rate [START_REF] Moreillon | Pathogenesis of streptococcal and staphylococcal endocarditis[END_REF][START_REF] Paz | A review of current treatment strategies for infective endocarditis[END_REF] . The pathogenesis of IE leads to the formation of vegetations on damaged heart valves, where adherent bacteria, platelets, immune cells and fibrin aggregate [START_REF] Holland | Infective endocarditis[END_REF] . S. aureus develops a biofilm inside vegetations with decreased metabolic activity [START_REF] Otto | Staphylococcal biofilms[END_REF] . Biofilm plays an important role in the difficulties encountered when treating staphylococcal endocarditis via three main Accepted manuscript effects: i) poor antibiotic penetration inside the biofilm, ii) decreased bacterial metabolic status leading to reduced antibiotic activity; iii) biofilm detachment leading to secondary localizations and/or thromboembolic events. As Staphylococci regularly colonize human skin, they are particularly involved in health-care associated IE, with a prevalence of both hospital and community-acquired methicillinresistant isolates (MRSA) reaching 43 % in some European countries in 2021 [START_REF]Surveillance Atlas of Infectious Diseases[END_REF] while communityacquired MRSA has become predominant in North America [START_REF] Kavanagh | Control of MSSA and MRSA in the United States: protocols, policies, risk adjustment and excuses[END_REF] .

Despite improvements in diagnosis, management tools and surgery, in-hospital mortality of patients with IE remains stable at 20% 1 . Current international guidelines recommend vancomycin (VAN) as the first-line therapy for MRSA native (monotherapy) and prosthetic valve (combination) IE [START_REF] Baddour | Infective Endocarditis in Adults: Diagnosis, Antimicrobial Therapy, and Management of Complications: A Scientific Statement for Healthcare Professionals From the American Heart Association[END_REF][START_REF] Habib | Infective Endocarditis: Epidemiology, Diagnosis, Imaging, Therapy, and Prevention[END_REF] . Long halflife lipoglycopeptides such as dalbavancin (DAL) could be of interest, with evidence of its antibiofilm activity in preclinical infection models [START_REF] Buzón-Martín | Dalbavancin for the Treatment of Prosthetic Joint Infections: A Narrative Review[END_REF] . However, DAL is not currently recommended for IE treatment.

Data evaluating the effect of DAL on a biofilm-producing infection and comparing its antimicrobial activity to that of vancomycin in an original model could provide additional arguments for its therapeutic use in the management of IE.

In addition, a model of this sort needs to highlight the bacterium-host-antibacterial interactions. Most of the current IE models use in vitro or animal supports. These approaches are complex to implement and raise ethical issues. We have therefore developed an ex-vivo model of human heart valve infection by S. aureus so as to more easily study bacteria-valve interactions, including the role of biofilm in IE, and also to consider bacterium-antibacterial interactions, in the presence of antibiotics.

Accepted manuscript

Material and methods

Heart valve infection model

Strains and conditions of bacterial growth

A clinical MRSA strain was isolated from blood cultures from a patient with proven health-careassociated aortic IE. Genome sequencing was performed using the MinION® device (Oxford Nanopore Technologies). Multilocus sequencing typing (MLST) and staphylococcal protein A (spa) typing were obtained from genome sequencing.

Human aortic heart valves were obtained from non-infected patients undergoing valve replacement surgery. Each heart valve collected was rinsed and cultured to ensure sample sterility. Freshly-removed heart valves were infected according to a protocol derived from Avilés-Reyes et al. [START_REF] Avilés-Reyes | Ex vivo Model of Human Aortic Valve Bacterial Colonization[END_REF] Briefly, the staphylococcal strain was grown in a brain-heart infusion broth (BHI, Oxoid®) at 37°C overnight. Noncalcified valves areas were sectioned with biopsy punch in 8-millimeter diameter sections. The sections were placed in 1mL of BHI containing a standardized inoculum at an optical density (OD) of 0.1, corresponding to a final inoculum of 5.10 7 colony-forming units (CFU) per mL. The valve sections were incubated for 48 hours at 37°C in brain-heart infusion broth (BHI, Oxoid®). The culture medium was renewed at 24h. Each valve section was rinsed 3 times in phosphate buffer solution to remove poorly adherent bacteria and thereafter sonicated with 500µL of Phosphate Buffer Saline (Sigma-Aldrich®) for 5 minutes (70kHz). The sonicated bacteria were plated to determine bacterial counts. The sterility of each heart valve was assessed by plating an uninfected control after 48 hours in sterile BHI.

Biofilm formation

Initial bacterial adhesion to heart valve sections was primarily assessed using Gram staining performed on paraffin-embedded histopathological sections of infected valve.

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Biofilm formation was assessed by labelling infected heart valve cryosections with anti-poly-N-β-(1-6)acetyl-glucosamine (PNAG) antibody, i.e. an exopolysaccharide involved in the maturation stage of the biofilm [START_REF] Lin | Elucidating the crucial role of poly N-acetylglucosamine from Staphylococcus aureus in cellular adhesion and pathogenesis[END_REF] . PNAG antibody was coupled with a secondary antibody (Alexa Fluor 488TM, Thermofischer® scientific) according to a protocol described by Maira-Litrán et al. [START_REF] Maira-Litrán | Synthesis and Evaluation of a Conjugate Vaccine Composed of Staphylococcus aureus Poly-N-Acetyl-Glucosamine and Clumping Factor A Manganelli R[END_REF] The labelled valve sections were analyzed by a fluorescence scanning system (NanoZoomer®, Hamamatsu). The presence of biofilm was confirmed by surface analysis of sections of infected valves by scanning electron microscopy (SEM), (Jeol®, Croissy Sur Seine, France). Valve sections previously infected and rinsed as described above were dehydrated in ethanol followed by critical point drying assay and sputter coated with gold palladium mixture.

Antimicrobial susceptibility testing MIC and MBC

DAL and VAN were provided by Sigma Aldrich. The minimum inhibitory concentration (MIC) was determined by the broth microdilution method according to CLSI guidelines for BHI broth with standardized inocula. Samples exhibiting no visible bacterial growth were seeded and incubated overnight. The minimum bactericidal concentration (MBC) was defined as the lowest concentration of antibiotic that prevented 99.9% of the growth of bacteria.

MbIC and MbEC

The VAN and DAL Minimum biofilm inhibitory concentrations (MbIC) and minimum biofilm eradication concentrations (MbEC) were determined by a modified Calgary device assay (called cMbIC/cMbEC) [START_REF] Ceri | The Calgary Biofilm Device: new technology for rapid determination of antibiotic susceptibilities of bacterial biofilms[END_REF] , by immersing pegs in a dedicated 96-well plate (MbEC Assay® Innovotech, Edmonton Canada) containing 150µL of OD 0.1 adjusted inoculum for 24h at 37°C. After rinsing in sterile water, the peg lids were incubated at 37°C for 24h in new wells containing 150 µL of twofold dilutions of DAL (with 0.002% polysorbate 80 supplementation) or VAN in BHI broth. The lowest antibiotic concentration for Accepted manuscript which there was no visible bacterial growth in well was identified as MbIC. Pegs exhibiting no visible bacterial in corresponding wells were rinsed and placed in a new 96-well plate containing BHI broth medium and then gently vortexed for 5 min at 70rpm for biofilm removal. MbEC was determined after 24h of incubation at 37°C.

MbIC and MbEC in heart valve sections

VAN and DAL MbIC and MbEC using heart valves sections (called vMbIC and vMbEC) were respectively determined on 2 mm heart valve sections in a standard 96-well plate by incubation in the same conditions as described above. Valve sections in wells exhibiting no visible bacterial growth were rinsed in sterile water thereafter sonicated for 5 min (70kHz) in a new 96-well plate containing BHI broth. vMbEC was determined after 24h of incubation at 37°C. Finally, heart valves were analyzed 48h after the onset of infection using SEM by adding DAL (with 0.002% polysorbate 80 supplementation) and VAN respectively at vMbEC after 24 h of incubation of the heart valve sections.

All experiments were performed at least in triplicate, the three most frequent identical results were retained. The study was approved by the Ethics Committee of Rennes (registration number 19.92).

Statistical analysis

Statistical analysis was performed with Prism 5.0® software (GraphPad Software, La Jolla, CA, USA)

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Results

Strain sequencing

Genome sequencing of staphylococcal isolates revealed clonal complex (CC) 8, sequence-type (ST) 8 spa type t008 strains, which was Panton-Valentine leucocidin negative. ST8 is the predominant S. aureus clone in Rennes university hospital. Pattern of antimicrobial resistance genes is presented in table S1.

Bacterial counts and biofilm formation

Bacterial counts on infected heart valves showed repeatable mean bacterial loads of 5.6±6. SEM analysis of the treated heart valves showed reduced staphylococcal colonies at vMbEC for DAL and VAN (see figure S2).

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Discussion

IE remains a hard to treat infection. As the low prevalence of IE makes clinical studies difficult to conduct, there is an urgent need for alternative models to improve patients management. Most of the knowledge regarding biofilm development in infective endocarditis has been provided from preclinical data, mainly on in vitro or animal models (rabbit and rat models) [START_REF] Lerche | Anti-biofilm Approach in Infective Endocarditis Exposes New Treatment Strategies for Improved Outcome[END_REF] . However, these models raise the issue of transposability to humans and may not strictly control all the different experimental conditions. Previous developments of ex vivo models of Enterococcus faecalis IE on a porcine valve compared to biofilm formation on an abiotic substrate demonstrated the limited relevancy of in vitro assays, given their ability to form biofilm on polystyrene for only half of the isolates, whereas all

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isolates formed biofilm on the ex vivo model [START_REF] Leuck | A widely used in vitro biofilm assay has questionable clinical significance for enterococcal endocarditis[END_REF] . These findings strengthen the need for alternative approaches, especially on human tissue to mimic human IE physiopathology in strictly controlled experimental conditions [START_REF] Magana | Options and Limitations in Clinical Investigation of Bacterial Biofilms[END_REF] .

Herein, we have established adhesion followed by biofilm production on human heart valves infected by S. aureus. The experimental conditions of our model are consistent with previous data indicating that the extracellular matrix of staphylococcal biofilms is completely formed after 6 to 9h of incubation 18 . In addition, as staphylococcal IE is associated with early valve damage, assessment of antistaphylococcal activity on a model mimicking IE conditions remains particularly challenging to better evaluate the efficacy of treatments for staphylococcal IE.

Only retrospective data on IE patients treated with DAL is available, with cure rates ranging from 72 to 91%, but DAL was usually used after several lines of treatment which had resulted in prior negativation of blood samples before the onset of DAL [START_REF] Lampejo | Dalbavancin and telavancin in the treatment of infective endocarditis: a literature review[END_REF] . Although both VAN and DAL inhibit bacterial cell wall synthesis, there are very few comparisons of their respective activity for IE management.

Previous in vitro studies of S. aureus biofilms have shown the superiority of DAL over VAN 20,[START_REF] Pilato | In vitro time-kill kinetics of dalbavancin against Staphylococcus spp. biofilms over prolonged exposure times[END_REF] . DAL has been compared in a single study to VAN in a rabbit model of staphylococcal IE and it showed greater efficacy than VAN 20 . Although our experimental conditions do not include either platelets and fibrin, which are involved in the formation of vegetation, nor the effect of the immune response, our results are consistent with this animal model, with a 28-fold increase in VAN MIC on a pre-established biofilm. Although our study is actually based on a single clinical isolate, these values confirm the interest of our easy-to-implement alternative model to study antibiofilm activity on human device.

Our results were also similar for both the Calgary device and our ex-vivo model. DAL showed strong antibiofilm activity with cMbIC <0.01 mg/L. This value is consistent with documented MbIC90 values of <1 mg/L in a recent review [START_REF] Buzón-Martín | Dalbavancin for the Treatment of Prosthetic Joint Infections: A Narrative Review[END_REF] . DAL showed comparable cMbEC for pegs to that for for heart valves (0.03 and 0.06 mg/L respectively). Thus, DAL showed very encouraging results on human biological device with strong activity, including on biofilm, and with very low vMbEC (0.03 mg/L). This value is

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largely achieved in plasma concentrations measured in humans [START_REF] Nicolau | Pharmacokinetics of dalbavancin in plasma and skin blister fluid[END_REF] . In addition to its potentially high level of activity, DAL nephrotoxicity could be less frequent than it is with VAN 23 . This characteristic could be an additional argument for its use in IE where a staphylococcal etiology has been associated

with an increased risk of renal failure, leading to additional difficulties in the management with VAN treatment [START_REF] Conlon | Predictors of prognosis and risk of acute renal failure in bacterial endocarditis[END_REF] . Taken together, our results on a human model confirm previous in vitro data and can support the use of DAL as an alternative to VAN in IE treatment induction or as output relay.

Conclusion

Dalbavancin showed promising antibiofilm activity compared to vancomycin in an original ex-vivo MRSA infective endocarditis model. Interestingly, dalbavancin remained highly bactericidal, including on biofilm, with very low MIC and MbEC values. Our results provide new evidence of DAL antibiofilm activity applied to IE, and this could provide further evidence for its potential in the medical management of this pathology.

  10 7 CFU/mL for the MRSA strain. Gram staining on histopathological sections of infected heart valves demonstrated peripheral bacterial adherence on damaged areas as shown in fig. 1. Wide-field fluorescence analysis of heart valve cryosections previously labelled with PNAG and its secondary fluorescent antibody revealed specific areas of fluorescence, overlapping with the presence of staphylococcal colonies revealed by the subsequent Gram staining (figS1). SEM analysis performed on 48-hour infected heart valve sections confirmed the presence of numerous colonies of aggregated staphylococci embedded in a matrix (fig.2), and confirmed the presence of a biofilm on damaged valve areas.

Figure 1 .

 1 Figure 1. Gram staining on histopathological sections of infected (A, B) and non-infected heart valves (C). Staphyloccocal colonies (arrows) are marginally adherent to valves. (Magnification, x20).

Figure 2 .

 2 Figure 2. SEM images of 48h infected heart valves sections (A) (magnification, x5,000) with production of staphylococcal biofilm on heart valve structure (B) (magnification, x10,000) and non-infected heart valve section (C) (magnification, x5,000).

Susceptibility testing and biofilm assays DAL

  and VAN MIC/MBC, cMbIC/cMbEC, vMbIC/ vMbEC are summarized in table 1.

Table 1 . Dalbavancin and vancomycin activity assessed on planktonic (MIC/MBC) and sessile conditions (MbIC/ MbEC) on Calgary device and heart valve sections.

 1

					S. aureus clinical strain
		planktonic		Calgary device		heart valve
		MIC	MBC	cMbIC	cMbEC	v MbIC	vMbEC
	DAL	<0.01	<0.01	<0.01	0.06	<0.01	0.03
	VAN	1	16	28	>64	28	32

MIC: minimum inhibitory concentration, MBC: minimum bactericidal concentration cMbIC: minimum biofilm inhibitory concentration on Calgary device, MbEC: minimum biofilm eradication concentration on Calgary device vMbIC: minimum biofilm inhibitory concentration on heart valves sections, MbEC: minimum biofilm eradication concentration on heart valves sections Values are expressed in mg/L.

Acknowledgments:

The authors thank Loïc Joanny for his expertise

Transparency declarations

No conflicts of interest to report

Funding

This work was supported by Inserm, UMR_S 1230 "Bacterial Regulatory RNAs and Medicine" Rennes 1 University, and Rennes clinical investigation center