Decompositions of packed words and self duality of Word Quasisymmetric Functions

By Foissy's work, the bidendriform structure of the Word Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual. However, the only known explicit isomorphism due to Vargas does not respect the bidendriform structure. This structure is entirely determined by so-called totally primitive elements (elements such that the two half-coproducts vanish). In this paper, we construct two bases indexed by two new combinatorial families called red (dual side) and blue (primal side) biplane forests in bijection with packed words. In those bases, primitive elements are indexed by biplane trees and totally primitive elements by a certain subset of trees. We carefully combine red and blue forests to get bicolored forests. A simple recoloring of the edges allows us to obtain the first explicit bidendriform automorphism of WQSym.


Introduction
Combinatorial Hopf algebras are a common meeting point of different communities. On one hand, the operad theory allows to construct them frequently since the free algebra on one generator admits a Hopf structure. For many operads, one can make the structure explicit using combinatorics, one of the most basic example being the free dendriform algebra on one generator realized as the Loday-Ronco hopf algebra of binary trees [1].
On the other hand, the theory of symmetric functions often proceeds through noncommmutative lifting to better understand the identities. Hence, the community introduced a series of larger and larger Hopf algebras over a large variety of combinatorial structures. One of the first step was the introduction of the dual pair of quasisymmetric functions and non-commutative symmetric functions [2,3] to understand the inner product of characters through the descent algebra [4]. It leads to the discovery of the Malvenuto-Reutenauer algebra FQSym of permutations. Another example was the introduction by Poirier [5] of an algebra of Young tableaux. In [6], it was realized that it can lead to a very simple proof of the Littlewood-Richardson rule.
An early meeting between the symmetric function and the operad communities was the discovery [7] that the same procedure allows to construct both the algebra of tableaux and the algebra of binary trees from the algebra of permutations. One just has to enforce some simple relation (respectively plactic and sylvester relation) in the variable of the polynomial realization.
Aside from the algebra of permutations, there is another non-commutative lifting of quasi-symmetric functions. Indeed Hivert's action on polynomials whose invariant are quasi-symmetric functions [6,8] can be lifted to words. Here, its non-commutative invariants spans the Hopf algebra WQSym of packed words or, equivalently, surjections or even ordered set partitions. This algebra has various applications in the theory of free Lie algebras is closely related to the Solomon-Tits algebra and twisted descents, the development of which was motivated by the geometry of Coxeter groups, the study of Markov chains on hyperplane arrangements (see [9] and the reference therein).
To better understand these algebra, one has to investigate their structure. For the binary tree algebra it was shown in [7] that it is free as an algebra and isomorphic to its dual. Though those properties are quite obvious for the algebra FQSym of permutation, the situation of WQSym is much more difficult. Its first study is due to Bergeron-Zabrocki [10]. They showed that it is free and co-free. However, it was only conjectured in [6] that its primitive Lie algebra is free and that it is self dual. It is only by a deep theorem of Foissy [11] that one can show that the second one is too. In particular, until Vargas's work [12], no concrete isomorphism was known.
Independently, Novelli-Thibon worked on parking functions which is a super-set of packed words. They endowed the hopf algebra of parking functions PQSym with a bidendriform bialgebra structure [13]. Then they describe WQSym as a sub-bidendriform bialgebra of PQSym [14]. Recall that a dendriform algebra is an abstraction of a shuffle algebra where the product is split in two half-products. If the coproduct is also split, and certain compatibilities hold, one gets the notion of bidendriform bialgebra [11].
Building on the work of Chapoton and Ronco [15,16], Foissy [11] showed that the structure of a bidendriform bialgebra is very rigid. In particular, he defined a specific subspace called the space of totally primitive elements, and showed that it characterizes the whole structure. This does not only re-prove the freeness and co-freeness, as well as the freeness of the primitive lie algebra, but also shows that the structure of a bidendriform bialgebra depends only on its Hilbert series (the series of dimensions of its homogeneous components). In particular, any such algebra is isomorphic to its dual. However, Foissy's isomorphism is not fully explicit and depends on a choice of a basis of the totally primitive elements. To this end, one needs an explicit basis of the totally primitive elements. Foissy described such a construction for FQSym [17]. In this paper, we construct a far reaching generalization for packed words and WQSym so that the basis discribed in [17] is simply a restriction to permutations and FQSym is a subbidendriform bialgebra of WQSym. We provide two explicit bases of totally primitive elements, for WQSym and its dual, using a bijection with certain families of trees called biplane.
We begin with a background section presenting Foissy's two rigidity structure theorems that prove, among other things, the self-duality of any bidendriform bialgebra (Theorem 2 and Corollary 3). We then define the notion of packed words as well as the two specific bases (Q and R) of WQSym and its dual, which will be the starting point of our combinatorial analysis.
Section 2 is devoted to the combinatorial construction of biplane forests (Definitions 67 and 102) which are our first key ingredient. They record a recursive decomposition of packed words according to their global descents (Lemma 16) and positions of the maximum letter (Lemma 42) or the value of the last letter (Lemma 81). We show that the cardinalities of some specific sets of biplane trees match the dimensions of primitive and totally primitive elements (Theorems 75 and 111).
In Section 3 we construct two new bases (O and P Definitions 116 and 124) of WQSym and its dual which each contain as a subset a basis for the primitive and totally primitve elements (see Theorems 118 and 126). To do so we decompose the space of totally primitive elements as a certain direct sum which matchs the combinatorial decomposition of packed words (Lemmas 115 and 123).
Finally in Section 4 we make explicit how bases O and P are sufficient to have an infinite number of bidendriform automorphism of WQSym. Then we give an explicit isomorphism based on an involution on packed words. The definition of the bijection require a new kind of forest mixing red and blue, namely bicolored-packed forests.

Cartier-Milnor-Moore theorems for Bidendriform bialgebras
The goal of this section is to recall the elements of the definition of bidendriform bialgebras which are useful for the comprehension of this paper. We refer to [11] for the full list of axioms.
A bialgebra is a vector space over a field K, endowed with an unitary associative product · and a counitary coassociative coproduct ∆ satisfying a compatibility relation called the Hopf relation ∆(a · b) = ∆(a) · ∆(b). In this paper all bialgebras are assumed to be graded and connected (i.e. the homogeneous component of degree 0 is K). They are therefore Hopf algebras, as the existence of the antipode is implied.
A bidendriform bialgebra is a K-vector space which is both a dendriform algebra and a co-dendriform co-algebra satisfying a set of four relations relating respectively ≺ and ≻ with ∆ ≺ , ∆ ≻ (see [11] for more details). In these equations, we use a kind of Einstein notation where∆(a) = a ′ ⊗ a ′′ and ∆ α ( 1.10) Adding those four relations shows that · and ∆ as defined above defines a proper bialgebra.
We recall here the relevant results of Foissy [11] on the rigidity of bidendriform bialgebras based on the works of Chapoton and Ronco [15,16].
Let A be a bidendriform bialgebra. We denote Prim(A) : = Ker(∆) the set of primitive elements of A. We also denote by A(z) and P(z) the Hilbert series of A and Prim(A) defined as A(z) : = ∑ +∞ n=1 dim(A n )z n and P(z) : = ∑ +∞ n=1 dim(Prim(A n ))z n . The present work is based on two analogues of the Cartier-Milnor-Moore theorems [11] which we present now. The first one is extracted from the proof of [17,Proposition 6]: Proposition 1. Let A be a bidendriform bialgebra and let p 1 . . . p n ∈ Prim(A). Then the map p 1 ⊗ p 2 ⊗ . . . ⊗ p n → p 1 ≺ (p 2 ≺ (. . . ≺ p n ) . . .).
( 1.11) is an isomorphism of co-algebras from T + (Prim(A)) (the non trivial part of the tensor algebra with deconcatenation as coproduct) to A. As a consequence, taking a basis (p i ) i∈I of Prim(A), the family (p w 1 ≺ (p w 2 ≺ (· · · ≺ p w n ) . . . )) w where w = w 1 . . . w n is a non empty word on I defines a basis of A. This implies the equality of Hilbert series A = P/(1 − P).
One can further analyze Prim(A) using the so-called totally primitive elements of A defined as TPrim(A) : = Ker(∆ ≺ ) ∩ Ker(∆ ≻ ). The associated Hilbert serie is defined as T (z) : = ∑ +∞ n=1 dim(TPrim(A n ))z n . Recall that a brace algebra is a K-vector space A together with an n-multilinear operation denoted as . . . for all n ≥ 2 which satisfies certain relations (see [15] for details).
Theorem 2 ([17,Theorem 4 and 5]). Let A be a bidendriform bialgebra. Then Prim(A) is freely generated as a brace algebra by TPrim(A) with brackets given by A basis of Prim(A) is described by ordered trees that are decorated with elements of TPrim(A) where p n is the root and p 1 , . . . , p n−1 are the children (see [15,16,17]). This is reflected on their Hilbert series as [11,Corollary 37]: T = A/(1 + A) 2 or equivalently P = T (1 + A).
Using Proposition 1 and Theorem 2 together with a dimension argument, one can show the two following corollaries: 17,Theorem 2]). Let A be a bidendriform bialgebra. Then A is freely generated as a dendriform algebra by TPrim(A).
Corollary 4 ([20, 21]). A basis of A is described by ordered forests of ordered trees that are decorated with a basis of TPrim(A).
On this basis, the product can be described using grafting (see Proposition 28 in [21]) and the coproduct as the deconcatenation of forests that are word of trees (see Theorem 35 equation 7.(c) in [20]).

Packed words
The algebra WQSym is a Hopf algebra whose bases are indexed by ordered set partitions or equivalently surjections or even packed words. In this paper, we use the latter which we define now.
In this paper we will deal with words over the alphabet of positive integers N >0 . We start with basic notations: First, max(w) is the maximum letter of the word w with the convention that max(ǫ) = 0. Then |w| is the length (or size) of the word w. The concatenation of the two words u and v is denoted as u · v. The shift of a word w of a value i is denoted by w is the left-shifted (resp. right-shifted) concatenation of the two words where all the letters of the left (resp. right) word are shifted by the maximum of the right (resp. left) word: 1121/3112 = 44543112 and 1121\3112 = 11215334. We also use the notation u| ≤i (resp. u| >i ) for the subword containing all letters smaller (resp. strictly greater) than a value i.

Definition 5.
A word over the alphabet N >0 is packed if all the letters from 1 to its maximum m appears at least once. By convention, the empty word ǫ is packed. For n ∈ N, we denote by PW n the set of all packed words of length (also called size) n and PW = n∈N PW n the set of all packed words.  Table 1: Number of packed words of size smaller than 9. Definition 6. The packed word u : = pack(w) associated with a word over the alphabet N >0 is obtained by the following process: if b 1 < b 2 < · · · < b r are the distinct letters occurring in w, then u is the image of w by the homomorphism b i → i.
A word u is packed if and only if pack(u) = u.
Example 7. The word 4152142 is not packed because the letter 3 does not appear while the maximum letter is 5 > 3. Meanwhile pack(4152142) = 3142132 is a packed word.

Definition 8.
The standardized word std(w) associated with a word over the alphabet N >0 is obtained by iteratively scanning w from left to right, and labelling the occurrences of its smallest letter, then labelling the occurrences of the next one, and so on. For the reader familiar with ordered set partitions, there is a classical bijection between packed words and ordered set partitions. The one corresponding to a packed word w 1 · w 2 · · · w n is obtained by placing the index i into the w i -th block. To depict some definitions or lemmas, we will use box diagrams with Cartesian coordonates for packed words. On these diagrams, positions are from left to right (as reading direction) and values are from bottom to top. These diagrams will also be used to represent different decompositions with different colors. Transparency will order the decompositions.
Example 11. Here we have three examples: the representation of the packed word 214313. Then the word 3415251 decomposed with red-factorization (see Lemma 42). Finally the general case of the red-blue-factorization (see Definition 135) where it can be seen clearly that the blue-factorization is done after the red-factorization thanks to transparency. Global descent are defined in [22] on permutations, here we generalise the definition on packed words.

Definition 12.
A global descent of a packed word w is a position c such that all the letters before or at position c are strictly greater than all letters after position c.
Example 13. The global descents of w = 54664312 are the positions 5 and 6. Indeed, all letters of 54664 are greater than the letters of 312 and this is also true for 546643 and 12.

Definition 14.
A packed word w is irreducible if it is non empty and it has no global descent.  Example 15. The word w ′ = 21331 is irreducible.

Lemma 16.
Each word w admits a unique factorization as w = w 1 /w 2 / . . . /w k such that w i is irreducible for all i.
Example 17. The global descent decomposition of 54664312 is 21331/1/12. The word n · n − 1 · ... · 1 has 1/1/.../1 as global descent decomposition. Definition 18. u ¡ v denotes the shuffle product of the two words. It is recursively defined by u ¡ ǫ : = ǫ ¡ u : = u and ( 1.12) where u and v are words and a and b are letters. Analogously to the shifted concatenation, one can define the right shifted-shuffle where all the letters of the right word v are shifted by the maximum of the left word u.

Definition 22.
u¡v denote the dualisation of the deconcatenation using the function pack(w) of Definition 6. u¡v 1.14) where u, v and u ′ · v ′ are packed words. We also use the non-overlapping shuffle product on values by adding the constraint that letters of the two parts are distinct: Example 23. 12¡11 = 1211 + 1222 + 1233 + 1322 + 2311, 12¢11 = 1233 + 1322 + 2311.
Remark 24. Using the classical bijection between ordered set partitions and packed words (see Example 10), the product ¢ is equivalent to the shifted shuffle on ordered set partitions defined in [10].

Definition 25.
Analogously to the two half shuffle product of Definition 20, we split the two products ¡ and ¢ in two parts.
( 1.16) u To sum up in a few words, in u ≺ v the last letter is comming from u and the rest is shuffled, in u v the maximum value is comming from u and the rest is shuffled.

The Hopf algebra of word-quasisymmetric functions WQSym
We are now in position to define the Hopf algebra of word-quasisymmetric functions WQSym. It was first defined as a Hopf algebra in [8]. Novelli-Thibon proved later that WQSym and its dual are bidendriform bialgebras [14,Theorems 2.5 and 2.6]. Their products and coproducts in the monomial basis (M w ) w∈PW involve overlapping-shuffle. However, to deal with the bidendriform structure, it will be easier for us to chose, among the various bases known in the literature [8,10,14,12] a basis where the shuffles are non-overlapping. Therefore, for WQSym * , we take the basis denoted (Q w ) w∈PW of [10,Equation 23] using the classical bijection between ordered set partitions and packed words (see Example 10). For the primal WQSym, we define the dual basis denoted (R w ) w∈PW . In this section we transfer the bidendriform structure on the bases (Q w ) w∈PW and (R w ) w∈PW .
Following Novelli-Thibon, we start from the basis (M w ) w∈PW and compute expressions of half product Equation (1.22) and half coproduct Equations ( 1.23) and (1.24) in the basis (Q w ) w∈PW . Then we dualise these operations Equations ( 1.25) to (1.27) to define the basis (R w ) w∈PW . The Hopf algebra product and reduced coproduct are respectively recovered as the sum of the half products and half coproducts.
The monomial word-quasisymmetric function of a totally ordered alphabet A associated to the packed word u is the linear combination of words defined by It turns out that the concatenation of two such elements is a sum of (M u ) u∈PW so that WQSym : = Vect(M u | u ∈ PW) is an algebra. This can be refined to a bidendriform bialgebra structure. The operations ≪, ≫, ∆ ≪ and ∆ ≫ on (WQSym) + : = Vect(M u | u ∈ PW n , n ≥ 1) are defined in the following way: for all u = u 1 · · · u n ∈ PW n≥1 and v ∈ PW m≥1 , ( 1.20) Example 27.
Theorem 28. [14,Theorem 2.5] As we said, we want a basis without overlapping-shuffle for the product. Following [10], we first define a partial order on packed words then we define the new basis.
Definition 29. [10] We say that the packed word u is smaller than v for the relation ≤ * if u and v have the same standardization and if u i = u j implies v i = v j for all i and j.
We give two immediate lemmas on this order that are useful. Lemma 31. For u ≤ * v, let i and i ′ such that u| ≤i and v| ≤i ′ are of the same size then u| ≤i ≤ * v| ≤i ′ and u| >i ≤ * v| >i ′ .
Proof. It is immediate with the definition of ≤ * .
Now we can recall ( [10,Equation 23]) the definition of the basis (Q w ) w∈PW Example 32.
It is proved in [10,Theorem 17] that the product in basis (Q w ) w∈PW is Thanks to Lemma 30, we have the two expressions for the two half products.
( 1.22) For the coproduct, we start with the definition of the coproduct in basis M.
Then, with Lemma 30 we have the two expressions for the two half coproducts.
Finally we define ≺, ≻, ∆ ≺ and ∆ ≻ on (WQSym * ) + : = Vect(R u | u ∈ PW n , n ≥ 1) by dualizing half products and half coproducts of the basis (Q w ) w∈PW in the following way: for all u = u 1 · · · u n ∈ PW n≥1 and v ∈ PW m≥1 , ( 1.27) Example 34. From now on Prim(WQSym) and TPrim(WQSym) are respectively abbreviated to Prim and TPrim. Moreover, we denote homogeneous components using indices and dualization using a * in exponent as in Prim * n . We give the first values of the dimensions a n : = dim(WQSym n ), p n : = dim(Prim n ) and t n : = dim(TPrim n ):  Though the numbers (t n ) n are easy to obtain thanks to the relation of Theorem 2: T = A/(1 + A) 2 , no combinatorial interpretation existed. The first results of this paper are two different subsets of packed words that are counted by these dimensions (redirreducible and blue-irreducible).

Decorated forests
In this paper we will generalize twice the construction of [17], one for WQSym and one for its dual. This section is devoted to the combinatorial ingredient, that is a notion of biplane forests suitable for indexing the various bases of primitive elements. Each time, we start by decomposing packed words through global descents and removal of specific letters. We then perform those decompositions recursively, encoding the result in a forests. We hence obtain so called biplane forests, which are in bijection with packed words. Later, the recursive structure of forests will be understood as a chaining of brace and dendriform operations generating some elements of WQSym or its dual. This will allow us to construct two bases of respectively TPrim and its dual by characterizing a subfamily of biplane trees.
From now on, we associate the color blue to the primal (WQSym) and the color red to the dual (WQSym * ). We start by explaining the construction on WQSym * (red) then we dualize the construction to the primal WQSym (blue).

Dual (Red)
For the red side WQSym * , the decomposition of packed words is made through global descents and removal of maximum values. One step of this decomposition is called the red-factorization.

Decomposition of packed words through maximums
In this section, we define two combinatorial operations on packed words (φ I and ◮) and the red-factorization that uses them. The unary operation φ I inserts new maximums in a word in positions I. A word that cannot be factorized u◮v in a non trivial way is called red-irreducible. Red-irreducible words will index our basis of TPrim * .
Definition 36. Fix n ∈ N and w ∈ PW n . We write m ′ : = max(w) + 1. For any p > 0 and any subset I ⊆ [1, . . . , n + p] of cardinality p, we define φ I (w) : = u 1 . . . u n+p as the packed word of length n + p obtained by inserting p occurrences of the letter m ′ in w so that they end up in positions i ∈ I. In other words u i = m ′ if i ∈ I and w is obtained from φ I (w) by removing all occurrences of m ′ . Notice that φ I (w) is only defined if n + p ≥ i p .
Notations 38. For the rest of this paper, I = [i 1 , . . . , i p ] will always denote a non-empty (p > 0) list of increasing non-zero integers. For any integer k, I ′ = I + k denote the list I ′ = [i 1 + k, . . . , i p + k]. Let PW I n denote the set of packed words of size n whose maximums are in positions i ∈ I. This way φ I (w) ∈ PW I n+p for any w ∈ PW n .
Lemma 39. Let n ∈ N and p > 0, for any I = [i 1 , . . . , i p ] ⊆ [1, . . . , n + p] of size p, φ I is a bijection from PW n to the PW I n+p . Moreover, for any W ∈ PW ℓ where ℓ > 0 there exists a unique pair (I, w) where I ⊆ [1 . . . ℓ] and w is packed, such that W = φ I (w).

The box diagram that pictures this lemma is
Proof. Let W ∈ PW ℓ with ℓ > 0 and m the value of the maximum letter of W. Let 1, . . . , ℓ] be the list of the positions of m in W and let w be the word obtain by removing all occurrences of m in W, then W = φ I (w). If φ I (u) = φ J (v) then positions of maximum values are the same so I = J and words obtain by removing these maximum values are also the same so u = v.
In other words, we remove the maximum letter of the right word, perform a left shifted concatenation and reinsert the removed letters as new maximums.
We call it the red-factorization of a word.
Example 43. Here is a first detailed example of a red-factorization of an irreducible packed word: Consider the irreducible packed word w = 543462161.
• The first step is to remove all the occurrences of the maximal value but keep in memory the positions in the initial word. We get w ′ = 5434_21_1 which is a packed word, but is not irreducible.
• The second step is to decompose the new word w ′ in irreducible factors w ′ = 1/ 212/_1/1_1. We still keep in memory the positions of the removed value. (when we have the choice, we cut to the left of the removed value.) • We can distinguish two groups of factors, those strictly before the first maximum withdrawn and the others w ′ = 1/212 / _1/1_1.
• Finally, by numbering the positions of the maximum removed value in the right factor (positions 1 and 4), we get the following decomposition of w (see Definition 36 for φ and Definition 40 for ◮): Example 44. Here are some other red-factorizations: the unique decomposition into irreducibles. Let ℓ be such that w ′ ℓ is the last factor which is entirely before the first removed maximum, it is the only choice to maximize the size of u. Then with r = k − ℓ we can rewrite w ′ as (u 1 / . . . /u ℓ )/(v 1 / . . . /v r ). Now we get I ′ by subtracting |u 1 / . . . /u ℓ | to all parts of I (I ′ = I − |u 1 / . . . /u ℓ |) and we obtain

Definition 45.
A packed word w is said to be red-irreducible if w is irreducible and the equality w = u◮v implies that u = ǫ(and w = v).
Here are all red-irreducible packed words of size 1, 2, 3 and 4 in lexicographic order: Here are some useful lemmas on the operation ◮.
Corollary 49. For any u, v ∈ PW, we have that u◮v is irreducible if and only if v is irreducible.
Proof. By contradiction, if v = v 1 /v 2 then by Lemma 47 u◮v = (u◮v 1 )/v 2 . Now if u◮v = w 1 /w 2 , as the position of the first maximum of u◮v is greater than |u| we have that Proposition 50. For any word w, w = u◮v is the red-factorization of w if and only if v is red-irreducible.
Proof. Let w ∈ PW and let u◮v be the red-factorization of w. Let v 1 and v 2 such that v = v 1 ◮v 2 , then (u/v 1 )◮v 2 = w by Lemma 46, but in the red-factorization the size of u is maximized so |(u/v 1 )| ≤ |u| and then we have that v 1 = ǫ so v is red-irreducible. Let w ∈ PW and let u and v such that w = u◮v and v is red-irreducible. By contradiction, suppose that there exists For the reader who is familiar with ordered set partitions, all the definitions in Section 2.1 can be easily written with these. However in Section 2.2 it is easier to do all the definitions on packed words and in Section 4 we must have the same object on both sides to explicit the isomorphism. So we decided to stick to packed words.

Red-forests from decomposed packed words using φ
We now apply recursively the red-factorization of the previous section to construct a bijection between packed words and a certain kind of trees that we now define.
Definition 51. An unlabeled biplane tree is an ordered tree (sometimes also called planar) whose children are organized in a pair of two (possibly empty) ordered forests, which we call the left and right forests, a forest being an ordered list of trees.
In the picture, we naturally draw the children of the left (resp. right) forest on the left (resp. right) of their father.
Example 52. The biplane trees , and are different. Indeed in the first case, the left forest contains two trees and the right forest is empty, in the second case both forests contain exactly one tree while in the third case the left forest is empty and the right contains two trees. Here is an example of a bigger biplane tree where the root has two trees in both left and right forests .

Definition 53.
A skeleton biplane tree is a biplane tree where no node has a right forest.
These skeleton biplane trees can also be seen as planar trees. In [17] we have planar trees recursively labeled by planar trees. Skeleton biplane trees are similar to these planar trees, we prefer to see them as biplane tree with no right forest in order to keep some constistency.
Definition 54. The size of a biplane tree is the number of node in the tree.
Remark 55. Biplane forests F (i.e. ordered list of biplane trees T) are counted by the sequence A001764 in OEIS [24] whose explicit formula is a(n) = ( 3n n )/(2n + 1). Biplane forests are in bijection with ternary trees. The bijection is the following, in a biplane forest a node has a first left child and a first right child and a right brother. A consequence is that unlabeled biplane trees are counted by the sequence A006013 in OEIS [24] whose explicit formula is a(n) = ( 3n+1 n )/(n + 1). Indeed, biplane trees are in bijection with pair of ternary trees. Here is an example of the ternary tree in bijection with the biplane forest constituted of one tree, the big biplane tree in Example 52  Remark 56. As we can see on OEIS [24], sequence A006013 which counts unlabeled biplane trees is the dimensions of the free L-algebra on one generator (see [25]). It would be interesting to investigate the link between L-algebras and bidendriform bialgebras using biplane trees.
In our construction we will deal with labeled biplane trees with colored edges. For a labeled biplane tree, we denote by Node R (x, f ℓ , f r ) the tree whose edges are colored in red, root is labeled by x and whose left (resp. right) forest is given by f ℓ (resp. f r ). We also denote by [t 1 , . . . t k ] a forest of k trees. The edge color (for now, only red) will play a role later in the paper.
We now apply recursively the global descent decomposition and the red-factorization of Lemmas 16 and 42. We obtain an algorithm which takes a packed word and returns a biplane forest where nodes are decorated by red-irreducible packed words: Definition 58. We now define two functions F Rske and T Rske . These functions transform respectively a packed word and an irreducible packed word into respectively a skeleton biplane forest and a skeleton biplane tree labeled by red-irreducible words. These functions are defined in a mutual recursive way as follows: • for any packed word w, let w 1 /w 2 / . . . /w k be the global descent decomposition of w, then F Rske (w) : .
We want to prove that the functions F Rske and T Rske are bijections. To do that we first define two functions that are the inverses.
Definition 61. We now define two functions F * Rske and T * Rske that transform respectively a red-skeleton forest and tree into packed words. These functions are defined in a mutual recursive way as follows: Lemma 62. The functions F Rske and F * Rske (resp. T Rske and T * Rske ) are two converse bijections between packed words and red-skeleton forests (resp. irreducible packed words and red-skeleton trees). That is to say F -1 Rske = F * Rske and T -1 Rske = T * Rske .
Proof. We start to prove that domain and codomain are as announced (see Items (a) and (b) bellow), then we prove that the functions F Rske and F * Rske (resp. T Rske and T * Rske ) are inverse to each other (see Items (c) and (d) bellow).
(a) By Definition 58, a forest (resp. tree) obtain by F Rske (resp. T Rske ) is a red-skeleton forest (resp. tree). Indeed, thanks to Proposition 50 nodes are labeled by red-irreducible words because a red-factorization is done and nodes have no right children.
(b) We prove by a mutual induction that F * Rske returns a packed word and that T * Rske returnes an irreducible packed word. Indeed, we do an induction on the size of the forest or the tree. Here is our induction hypothesis for n ∈ N: ∀t, red-skeleton tree of size ≤ n, T * Rske (t) is an irreducible packed word, ∀ f , red-skeleton forest of size ≤ n, F * Rske ( f ) is a packed word.
( 2.1) The base case (n = 0) is given by the first item of Definition 61 (i.e. F * Rske ([]) = ǫ the empty packed word). Now, let us fix n ≥ 1 and suppose that the hypothesis (2.1) holds. Let f = [t 1 , . . . , t k ] be a red-skeleton forest of size n + 1. • If k = 1, then f is reduced to a single tree t. We need to prove that T * Rske (t) is an irreducible packed word (which also gives that F * Rske ( f ) is a packed word as in this case be a red-skeleton tree of size n + 1 (notice that the word v can be of any size). The size of f ℓ is n, so by induction F * , the forest contains at least two trees, since all trees are of size at least one, t 1 , . . . , t k are at must of size n, so we have by induction that T * and thus it is a packed word.
(c) We now prove by a mutual induction on the size of the forest or the tree that, for n ∈ N: The base case (n = 0) is given by the first item of Definitions 58 and 61 as F Rske (F * Rske ([])) = F Rske (ǫ) = []. Now let us fix n ≥ 1 and suppose that the hypothesis (2.2) holds. Let f = [t 1 , . . . , t k ] be a red-skeleton forest of size n + 1. • If k = 1, then the forest f is reduced to a single tree t, then it is sufficient to prove T Rske (T * Rske (t)) = t (as in this case F * a red-skeleton tree of size n + 1. As the label v is a red-irreducible packed word, with the induction hypothesis on F * Rske ( f ℓ ) and with Proposition 50, F * Rske ( f ℓ )◮v is the redfactorization so: • If k ≥ 2, since all trees are of size at least one, they are at most of size n, so we have by induction that: (d) Finally we prove by a mutual induction on the size of the word w that, for n ∈ N: ( 2.3) The base case (n = 0) is given by the first item of Definitions 58 and 61 as F * Rske (F Rske (ǫ)) = F * Rske ([]) = ǫ. Now let us fix n ≥ 1 and suppose that the hypothesis (2.3) holds. Let w ∈ PW n+1 a packed word of size n + 1.
Let w = u◮v be the red-factorization of w, then we can use the induction hypothesis on u, indeed as v is not empty the size of u is smaller than n: • If k ≥ 2, then we use the induction hypothesis on each factors, so we have: Now that we have the red-skeleton, we will add right forests to every nodes to obtain biplane trees. For every node, if v is the red-irreducible word in label, with φ I (v ′ ) = v, then I is the new label and F R (v ′ ) is the new right forest.
Here is the formal definition of F R (w) and T R (w) which are very similar to Definition 58, only the third item is different. The labels are now lists of integers.
Definition 63. The forest F R (w) (resp. tree T R (w)) associated to a packed word (resp. irreducible packed word) w are defined in a mutual recursive way as follows: • for any packed word w, let .
• for any irreducible packed word w, we define T R (w) .
Example 64. Consider again w = 876795343912. We start from the red-skeleton forest from Example 59.  . . . , ℓ g ] and f r = [r 1 , . . . , r d ], which is depicted as follows: In particular, if t is a single node then ω(t) = p. By extension, the weight of a forest is the sum of the weight of its trees.
Lemma 66. The weight of a forest (resp. a tree) obtained by the functions F R (resp. T R ) is equal to the size of the word, i.e. For allw ∈ PW then ω(F R (w)) = |w| and for all w ∈ PW with w irreducible then ω(T R (w)) = |w|. Proof. We prove by induction with the following hypothesis, for n ∈ N: ∀w ∈ PW n , ω(F R (w)) = |w|, ∀w ∈ PW n with w irreducible, ω(T R (w)) = |w|. (2.4) The base case is given by the first item of Definition 63 as F R (ǫ) = [] and ω([]) = |ǫ| = 0. Let us fix n ≥ 1 and suppose that the hypothesis (2.4) holds. Let w ∈ PW n+1 and w = w 1 /w 2 / . . . /w k be the global descent decomposition of w.
As p > 0, the sizes of u and v are at most n, by induction ω(T R (w)) = p + |u| + |v| = |w|.
• If k ≥ 2, by induction on each factors, we have that Definition 67. Using the same notations as in previous Definition 65, we say that t is a red-packed tree if it satisfies: . . , ℓ g and r 1 , . . . , r d are red-packed trees.
An ordered list of red-packed trees is a red-packed forest.
Remark 68. Red-skeleton trees can be interpreted as flattened representations of redpacked trees. Symmetrically, red-packed trees can be interpreted as unfolded representations of red-skeleton trees. We use the operation φ I to change between red-packed and red-skeleton trees.
Notations 69. From now on, we use these notations: • F Rn the set of red-packed forests of weight n, (F Rn : = {F R (w)} w∈PW n ), • T Rn the set of red-packed trees of weight n, (T Rn • N Rn the set of red-packed trees of weight n such that the left forest of the root is empty, (N Rn : = {T R (w)} w∈PW n with w red-irreducible). In particular, the redskeleton of a tree of N Rn consist of a single node labeled by a red-irreducible word.
Remark 70. The set N Rn can be described as a disjointed union of sets depending on I = [i 1 , . . . , i p ]. Let F I Rn denote the set of red-packed forests of weight n that can be right children of a node labeled by I (see Definition 67 for conditions), we have the following description: Analogously, we use F Rske , T Rske and N Rske for red-skeleton forests, trees and trees with only one node. We can remark that for n = 1 we have N R1 = T R1 = F R1 and ∀n > 1, N Rn T Rn F Rn .
As with Definition 61, we want to prove that the functions F R and T R are bijections. To do that we first define the two inverse functions.
Definition 71. We define here the functions F * R (resp. T * R ) that transform a red-packed forest f (resp. tree t) into a packed word. We reverse all instructions of Definition 63 as follows: • for any non empty red-packed forest • for any non empty red-packed tree t = Node R (I, f ℓ , f r ), then T * There might be a problem with this definition since φ We prove in the following Lemma 72 that the inequality holds if t ∈ T R .

Lemma 72.
For any red-packed forest f , F * R ( f ) is a well defined word of size ω( f ). For any red-packed tree t, T * R (t) is a well defined word of size ω(t). Proof. We prove by induction with the following hypothesis, for n ∈ N: ( 2.6) The base case is given by the first item of Definition 71 as F * R ([]) = ǫ and ω([]) = |ǫ| = 0. Let us fix n ≥ 1 and suppose that the hypothesis (2.6) holds. Let f = [t 1 , . . . , t k ] ∈ F Rn+1 .
• If k = 1, it is sufficient to prove the second item of (2.6). Let t = Node R (I, f ℓ , f r ) ∈ T Rn+1 . According to Definition 67 with notations of Definition 65 there are two cases: d = 0 and I = [1, . . . , p]. We have that i p = p and | F * R ( f r )| = |ǫ| = 0 so i p ≤ 0 + p and φ I (ǫ) = 11 . . . 11 of size p. Now by induction on f ℓ , we have that F As p > 0 we can apply the hypothesis (2.6) on f r and f ℓ . According to Definition 67 we have that • If k ≥ 2, the weight of trees are at least 1 so we can apply (2.6) on trees of f . Theorem 73. The functions F R and F * R (resp. T R and T * R ) are two converse bijections between packed words of size n and red-packed forests (resp. irreducible packed words and red-packed trees) of weight n. That is to say F -1 R = F * R and T -1 R = T * R .
Proof. The proof is very similar to the one of Lemma 62. Indeed, we start to prove that domain and codomain are as announced (see Items (a) and (b) bellow), then we prove that the functions F R and F * R (resp. T R and T * R ) are inverse to each other (see Items (c) and (d) bellow).
We now give the differences with the proof of Lemma 62 and we advise the reader to read the two proofs in parallel. While for Item (a) in Lemma 62 it was simple, we need to do an induction here to prove that conditions on labels are respected. For Items (b), (c) and (d), the same inductions are done with one additional argument, so only the different argument of the induction is explicited here.
(a) We prove by a mutual induction that F R returns a red-packed forest and that T R returns a red-packed tree. Indeed, we do an induction on the size of the word w. Here is our induction hypothesis for n ∈ N: ∀v ∈ PW, irreducible packed word of size ≤ n, T R (v) is a red-packed tree, ∀w, packed word of size ≤ n, F R (w) is a red-packed forest.
( 2.7) The base case (n = 0) is given by the first item of Definition 63. Now let us fix n ≥ 1 and suppose that the hypothesis (2.7) holds. Let w ∈ PW be a packed word of size n + 1 and let w 1 / · · · /w k be the global descent decomposition of w.
• If k = 1 (w is irreducible), then F R (w) is reduced to a single tree T R (w). We need to prove that T R (w) is a red-packed tree (which also gives that F R (w) is a red-packed forest). Let w = u◮v be the red-factorization of w.
The inequalities on I and v ′ in Lemma 42 are the same as the inequalities on I and f r in Definition 67. Therefore by Lemma 66 and (2.7) on f r , we have that T R (w) belongs to T R .
• If k ≥ 2, the hypothesis (2.7) can be applied to each factors.
(b) Compared to the proof of Lemma 62 we use the same general arguments to prove that F * R and F * Rske return a packed word. First of all the base case and the case were the size of the forest is k ≥ 2 are dealt with by a similar argumentation. It remains to prove that T * R returns an irreducible packed word. We thus suppose that the induction hypothesis (2.8) holds for a given n ∈ N: ∀t, red-packed tree of size ≤ n, T * R (t) is an irreducible packed word, ∀ f , red-packed forest of size ≤ n, F * R ( f ) is a packed word.
( 2.8) Let t = Node R (I, f ℓ , f r ) be a red-packed tree of size n + 1. By induction we have that F * R ( f ℓ ) and F * R ( f r ) are packed words. Moreover either f r = ǫ and I = [1, . . . , p] . . . , i p ] and f r = [r 1 , . . . , r d ]. In both cases, φ I (F * R ( f r )) is an red-irreducible packed word. Indeed, we recognize the two cases of Definition 45 with the same inequalities. Finally T * R (t) = F * R ( f ℓ )◮φ I (F * R ( f r )) is an irreducible packed word according to Corollary 49.
(c) We now want to prove that for any forest f (resp. t), F R (F * R ( f )) = f (resp. T R (T * R (t)) = t). As in Item (b), the arguments are the same as is the proof of Lemma 62 for F R and F Rske . In the case of T R the new arguments are the same as in Item (b) (i.e. φ I (F * R ( f r )) is a red-irreducible packed word). (d) Finally, we want to prove that for any packed word w (resp. irreducible packed word v), F * R (F R (w)) = w (resp. T * R (T R (v)) = v). Once again, the only difference with Lemma 62 is the former second case. It remains to prove that point and we thus suppose that the induction hypothesis (2.9) holds for n ∈ N: ∀v ∈ PW, irreducible packed word of size ≤ n, T * R (T R (v)) = v, ∀w ∈ PW, packed word of size ≤ n, F * R (F R (w)) = w.
( 2.9) Let v be an irreducible packed word of size n + 1. Let v = v ′ ◮φ I (v ′′ ) be the redfactorization of v. We have by Definition 63 that T R (v) = Node R (I, F R (v ′ ), F R (v ′′ )). As |I| > 0, the sizes of v ′ and v ′′ are smaller than n so we can apply (2.9). We have: Example 74. There is a unique forest in F R1 , namely 1 , here are the 3 forests of F R2 with the associated packed word:F R (12) = More examples can be found in the annexes section with Tables 7 to 12.
We conclude by the main theorem of this subsection. It is a generalization of the construction of [17] for FQSym and permutations to WQSym and packed words. Indeed, if we restrict the construction on permutations and we consider right children of a node as label of this node, we have the same construction as in [17] with a shift of 1 for labels.
Here are some examples of trees in [17] and the equivalent red-packed tree.
All the constructions with red-packed forests have been done in order to have this theorem.
Theorem 75. For all n ∈ N we have the three following equalities : dim(WQSym * n ) = #F Rn and dim(Prim * n ) = #T Rn and dim(TPrim * n ) = #N Rn Proof. Theorem 73 proves the first equality. It also gives a relation between #F R and #T R . Indeed a red-packed forest of weight n is an ordered sequence of red-packed trees of weight (n k ) such that ∑ k (n k ) = n. This relation is the same between dim(WQSym * n ) and dim(Prim * n ) according to Proposition 1 (i.e. A = P/(1 − P)). Red-skeleton trees are equivalent to ordered trees decorated by red-irreducible words as said in Remark 68. Recall that a basis of primitive elements is given by Theorem 2 as ordered trees decorated by totally primitive elements. Elements of N R are by definition in bijections with red-irreducible words, labels of red-skeleton trees.

Primal (Blue)
Now we do the same work for the primal side: WQSym. This subsection follows the same structure of statements as the previous one. Recall that in Section 2.1 we constructed a bijection between packed words and red-forests by recursively decomposing packed words using global descent and removal of maximums. In this section we follow the same path: incerting the last letter using ψ i α (lowercase i designates the integer value) instead of new maximums using φ I (uppercase I designates the list of their positions). We define a blue-factorization of packed words. When used recursively, bluefactorization and global descent decomposition construct a bijection between PW and so-called blue-packed forest. Since the general structure of proofs are the same as in the previous section, we will mostly focus on the differences between combinatorials arguments.

Decomposition of packed words through last letter
In this section, we define two combinatorial operations on packed words (ψ i α and ) and the blue-factorization that use them. The unary operation ψ i α insert the new value i at the end of a given word. A word that cannot be factorized u v in a non trivial way is called blue-irreducible. Blue-irreducible words will index a new basis of TPrim.
Definition 76. Fix n ∈ N and w ∈ PW n . For any 1 ≤ i ≤ max(w) + 1 (with the convention max(ǫ) = 0), we denote by ψ i • (w) = u 1 · · · u n · i the packed word defined by u k = w k if w k < i and u k = w k + 1 otherwise. We also define ψ i • (w) = w · i for any 1 ≤ i ≤ max(w).
Depending on α, the box diagram can be represented as . In the general case, we will note W = i α w .
Proof. Let W ∈ PW ℓ with ℓ > 0 and i the value of the last letter of W.
• If i appears multiple times in W, then let w = W 1 . . . W ℓ−1 , we only remove the last letter i of W. We have W = ψ i • (w).
• Otherwise, i appears only as the last letter, then let w = pack(W 1 . . . W ℓ−1 ), we remove the last letter i of W and pack the word. We have W = ψ i • (w).
If ψ i α (u) = ψ j β (v) then the last letter is the same so i = j, the multiplicity of this letter is the same so α = β and the prefix are the same u = v.
In other words, we remove the last letter of the right word, perform a reversed left shifted concatenation and adding back the last letter also shifted.
We call it the blue-factorization of a word.
Example 82. Here is a first detailled example of a blue-factorization of an irreducible packed word: Consider the irreducible packed word w = 654623314.
• The first step is to remove the last letter i = 4. Here there are multiple occurences of the last letter in w, then α = •, we get w ′ = 65462331 which is a packed word, but is not irreducible.
• The second step is to set w ′ 1 as the first irreducible factor of w ′ and u the rest of w ′ . This way w ′ = w ′ 1 /u and the size of u is maximized. Here w ′ 1 = 3213 and u = 2331.
• Finally, we get the following decomposition of w (see Definition 76 for ψ and Definition 79 for ): Here are some other blue-factorizations: Proof. Let w be irreducible and let (i, α, w ′ ) be the unique triplet such that w = ψ i α (w ′ ) according to Lemma 78. If i = max(w) and it appears only one time (i.e. α = • and i = max(w ′ ) + 1) then the blue-factorizations is w = w ′ ψ 1 • (ǫ).
In any other case, we write w ′ = w ′ 1 /w ′ 2 / . . . /w ′ k , the decomposition into irreducibles. Let u = w ′ 2 / . . . /w ′ k and i ′ = i − max(u). We have that i ′ ≤ max(w ′ 1 ) otherwise w wouldn't be packed and 1 ≤ i ′ otherwise w wouldn't be irreducible. If α = • then 1 = i ′ otherwise w wouldn't be irreducible. Then we have w = u ψ i ′α (w ′ 1 ) where the size of u is maximized.

Definition 85.
A packed word w is blue-irreducible if w is irreducible and w = u v implies that u = ǫ(and w = v).
Here are some useful lemmas on the operation . There are some similarities with Lemmas 46 and 47, Corollary 49, and Proposition 50.
Proof. Let u, v, w ∈ PW with w = ǫ, and let w ′ and i α such that w = ψ i α (w ′ ).
Remark 88. Theses relations are the same up to symmetry as the one with ◮ (Lemmas 46 and 47). So adding the associativity of shifted concatenation u/(v/w) = (u/v)/w, the two operations and / verify relations of the skew-duplicial operad [23].

Corollary 89. For any u, v ∈ PW, we have that u v is irreducible if and only if v is irreducible.
Proof. By contradiction, if v = v 1 /v 2 then by Lemma 87 u v = v 1 /(u v 2 ). Now if u v = w 1 /w 2 as the value of the last letter of u v is greater than max(u) we have that Proposition 90. For any word w, w = u v is the blue-factorization of w if and only if v is blue-irreducible. Proof. Let w ∈ PW and let u v be the blue-factorization of w. Let v 1 and v 2 such that v = v 1 v 2 , then (v 1 /u) v 2 = w by Lemma 86, but in the blue-factorization the size of u is maximized so |(v 1 /u)| ≤ |u| and then we have that v 1 = ǫ so v is blue-irreducible. Let w ∈ PW and let u and v such that w = u v and v is blue-irreducible. By contradiction, suppose that there exists u Thanks Remark 84 the following proposition is immediate.

Proposition 91.
A permutation σ is blue-irreducible if and only if σ −1 is red-irreducible.

Blue-forests from decomposed packed words using ψ
As in Section 2.1.2 we will apply recursively the blue-factorization of the former section to construct a bijection between packed words and a certain kind of labeled biplane trees.
In this construction, the labels can be a blue-irreducible word for skeleton, or an integer with a sign α ∈ {•, •}. In order to differentiate the trees from the one of the previous section, we will draw them in blue. As before, for a labeled biplane tree, we denote the trees by Node B (x, f ℓ , f r ). We apply recursively the global descent decomposition and the blue-factorization of Lemma 81. We obtain an algorithm which takes a packed word and returns a biplane forest where nodes are decorated by blue-irreducible words: Definition 93. Exactly as Definition 58 of F Rske and T Rske , we now define two functions F Bske and T Bske . These functions transform respectively a packed word and an irreducible packed word into respectively a biplane forest and a biplane tree. These functions are defined in a mutual recursive way as follow: • for any packed word w, let w 1 /w 2 / . . . /w k be the global descent decomposition of w, then F Bske (w) : = [T Bske (w k ), T Bske (w k−1 ), . . . , T Bske (w 1 )] (notice the inversion compared to Definition 58). .

Definition 95. A labeled biplane forest (resp. tree) is a blue-skeleton forest (resp. tree)
if and only if it is labeled by blue-irreducible words and no node has a right child.
We want to prove that the functions F Bske and T Bske are bijections. To do that, as for F Rske and T Rske , we first define the two inverse functions.
Definition 96. We now define two functions F * Bske and T * Bske that transform respectively a blue-skeleton forest and tree into packed words. These functions are defined in a mutual recursive way as follow: • for any blue-skeleton forest f = [t 1 , . . . , t k ], we define F * Bske ( f ) : = T * Bske (t k )/ . . . / T * Bske (t 1 ). (notice the inversion compared to Definition 61) Lemma 97. The functions F Bske and F * Bske (resp. T Bske and T * Bske ) are two converse bijections between packed words and blue-skeleton forests (resp. irreducible packed words and blue-skeleton trees). That is to say F -1 Bske = F * Bske and T -1 Bske = T * Bske . Proof. The proof structure is the same as the one of Lemma 62 with use of statements comming from this subsection. We can see in this Now that we have the blue-skeleton, we will add right forests to every nodes to have biplane trees. For every nodes, if v is the blue-irreducible word in the label, then with Here is the formal definition of F B (w) and T B (w) which is very similar to Definition 93, only the third item is different. The labels are now pairs of an integer and a sign α ∈ {•, •}. Definition 98. The forest F B (w) (resp. tree T B (w)) associated to a packed word (resp. irreducible packed word) w are defined in a mutual recursive way as follows: • for any packed word w, let w 1 /w 2 / . . . /w k be the global descent decomposition of w, then F B (w) : = [T B (w k ), T B (w k−1 ), . . . , T B (w 1 )] (notice the inversion compared to Definition 63). .
• for any irreducible packed word w, we define T B (w) .
Example 99. Consider again w = 8967647523314. We start from the blue-skeleton forest from Example 94.
F Bske (8967647523314) = 3431421 1 122 Definition 100. Let t be a labeled biplane tree. We write . . . , ℓ g ], and f r = [r 1 , . . . , r d ], which is depicted as follows: The weight of t (ω(t)) is the number of nodes with • in t. By extension, the weight of a forest is the sum of the weight of its trees.
Lemma 101. The weight of a forest (resp. a tree) obtain by the functions F B (resp. T B ) is equal to the maximum value of the word. i.e. ∀w ∈ PW, ω(F B (w)) = max(w), ∀w ∈ PW with w irreducible, ω(T B (w)) = max(w).

Proof.
We prove by induction with the following hypothesis, for n ∈ N: ( 2.10) The base case is given by the first item of Definition 98 as F B (ǫ) = [] and ω([]) = max(ǫ) = 0 by convention. Let us fix n ≥ 1 and suppose that the hypothesis (2.10) holds. Let w ∈ PW n+1 and w = w 1 /w 2 / . . . /w k be the global descent decomposition of w.
and α ∈ {•, •} be the blue-factorization of w, then, depending of α the node is counted or not: The sizes of u and v are at most n so by induction ω(T B (w)) = (1+) max(u) + max(v) = max(w).
• If k ≥ 2, by induction on each factors, we have that Definition 102. Using the same notations as in previous Definition 100, we say that t is a blue-packed tree if it satisfies: . . , ℓ g and r 1 are blue-packed trees.
(2.11) An ordered list of blue-packed trees is a blue-packed forest.
Remark 103. The same remark as Remark 68 can be done with blue-skeleton trees that can be interpreted as flattened representations of blue-packed trees. Symmetrically, bluepacked trees can be interpreted as unfolded representations of blue-skeleton trees. We use the operation ψ i α to change between blue-packed and blue-skeleton trees.
Notations 104. In the same way as Notations 69 we add the following notations: • F Bn the set of blue-packed forests of size n, (F Bn : = {F B (w)} w∈PW n ), • T Bn the set of blue-packed trees of size n, (T Bn : = {T B (w)} w∈PW n with w irreducible), • N Bn the set of blue-packed trees of size n such that the left forest of the root is empty, (N Bn : = {T B (w)} w∈PW n with w blue-irreducible). In particular, the blueskeleton of a tree of N Bn consist of a single node labeled by a blue-irreducible word.
Remark 105. The set N Bn can be described as a disjointed union of sets depending on i and α. Let F i α Bn denote the set of blue-packed forests of weight n that can be right children of a node labeled by i α (see Definition 102 for conditions), we have the following description: Analogously, we use F Bske , T Bske and N Bske for blue-skeleton forests, trees and trees with only one node. We can remark that for n = 1 we have N B1 = T B1 = F B1 and ∀n > 1, N Bn T Bn F Bn .
Once again we define two functions in order to prove that F B and T B are bijections.
Definition 106. We define here the functions F * B (resp. T * B ) that transform blue-packed forest f (resp. tree t) into a packed word. We reverse all instructions of Definition 98 as follows: • for any non empty blue-packed forest • for any non empty blue-packed tree t = Node B (i α , f ℓ , f r ), then T * 76), there might be a problem with this definition. We prove in the following Lemma 107 that this is the case if t ∈ T B .

Lemma 107.
For any blue-packed forest f , F * B ( f ) is a well defined word and its maximum value is ω( f ). For any blue-packed tree t, T * B (t) is a well defined word and its maximum is ω(t).

Proof.
We prove by induction with the following hypothesis, for n ∈ N: ( 2.13) The base case is given by the first item of Definition 106 as F * B ([]) = ǫ and ω([]) = max(ǫ) = 0. Let us fix n ≥ 1 and suppose that the hypothesis (2.13) holds. Let f = [t 1 , . . . , t k ] ∈ F Bn+1 . • If k = 1, it is sufficient to prove the second item of (2.13). Let t = Node B (i α , f ℓ , f r ) ∈ T Bn+1 . According to Definition 102 with notations of Definition 100 there are two cases: d = 0 and i α = 1 • . We have that max(F * B ( f r )) = max(ǫ) = 0 so i ≤ 0 + 1 and ψ i α (ǫ) = 1. Now by induction on f ℓ , we have that F * B ( f ℓ ) is a well defined word and its maximum value is ω( f ℓ ). Finally T * is a well defined word and its maximum is the last value: max(F * B ( f ℓ )) + 1 = ω(t). d = 1. In this case, we can directly apply the hypothesis (2.13) on f r and f ℓ . According to Definition 102 we have that • If k ≥ 2, the weight of trees are at least 1 so we can apply (2.13) on trees of f . Theorem 108. The functions F B and F * B (resp. T B and T * B ) are two converse bijections between packed words of size n and blue-packed forests (resp. irreducible packed words and blue-packed trees) of size n. That is to say F -1 B = F * B and T -1 Proof. The proof structure is the same as the one of Theorem 73 which is similar to the one of Lemmas 62 and 97. But with use of statements comming from this subsection.

Definition 67
Definition 102 red-packed forest and blue-packed forest. Definition 71 Definition 106 F * R , T * R and F * B , T * B .
Remark 109. As we can see in Sections 2.1 and 2.2, the role of size and weight are exchanged for red and blue-forests. For red forests, the size (number of nodes) is equal to the maximum letter of the word associated while the weight (Definition 65) is the number of letter of the associated word. For blue-forests, it is the opposite, the number of letters of the associated word is equal to the size of the forest while le maximum letter is equal to the weight (Definition 100) of the forest. That is why we denote the set of red packed forests of weight n by F Rn and the set of blue-packed forests of size n by F Bn .
Example 110. There is a unique forest in F B1 , namely 1 • , here are the 3 forests of F B2 with the associated packed word: We show below the forests of F B3 : More examples can be found in the annexes section with Tables 7 to 12. We conclude by the main theorem of this subsection. It is the dual of Theorem 75.
Theorem 111. For all n ∈ N we have the three following equalities : dim(WQSym n ) = #F Bn and dim(Prim n ) = #T Bn and dim(TPrim n ) = #N Bn Proof. The proof is similar to the one of Theorem 75 thanks to Theorem 108 instead of Theorem 73.

Bases for totally primitive elements
In this section we construct two bases of primitive and totally primitive elements of WQSym and WQSym * . Thanks to Theorems 75 and 111 we now have the combinatorial objects to index those bases and we know that their numbers agree with the dimensions. We therefore only need to show that they are linearly independent. We will proceed by showing that the decompositions through maximum and through last letter preserve the total primitivity. As in Section 2, we start by working on WQSym * associated to the color red and do the same work on the primal WQSym associated to the color blue.

Decomposition through maximums and totally primitive elements
We define a linear map Φ I : WQSym * → WQSym * as follows: for all n ∈ N and w = w 1 · w 2 · · · w n ∈ PW n , We define a projector τ I : WQSym * → WQSym * as follows: for all n ∈ N and w = w 1 · w 2 · · · w n ∈ PW n , if and only if i ∈ I, 0 else. (3.2) These are orthogonal projectors in the sense that τ 2 I = τ I and τ I • τ J = 0 (I = J).

Lemma 114.
For any I, we have Im(Φ I ) = Im(τ I ) where Im( f ) denotes the image of f .
Proof. For any I, the inclusion Im(Φ I ) ⊂ Im(τ I ) is automatic by definition of Φ I and τ I . Indeed, for any w ∈ PW n if i p ≤ n + p then Φ I (R w ) = R φ I (w) and τ I (R φ I (w) ) = R φ I (w) and Φ I (R w ) = 0 otherwise. By linearity Im(Φ I ) ⊂ Im(τ I ).
For any I, the inclusion Im(Φ I ) ⊃ Im(τ I ) is a consequence of Lemma 39 and linearity. Indeed, for any w ∈ PW, τ I (R w ) = R w ⇔ (w i = max(w) ⇔ i ∈ I). If τ I (R w ) = R w let w ′ be such that φ I (w ′ ) = w using Lemma 39, then Φ I (R w ′ ) = R w = τ I (R w ). By linearity Im(Φ I ) ⊃ Im(τ I ).

The new basis P
Definition 116. Let t 1 , . . . , t k be k red-packed trees, I = [i 1 , . . . , i p ], f l = [ℓ 1 , . . . , ℓ g ] be a red-packed forest and f r ∈ F I R be a red-packed forest that can be right children of a node labeled by I, Example 117.
More examples can be found in the annexes section with Table 13 and Figures 6 and 8.
Theorem 118. For all n ∈ N >0 1. (P f ) f ∈F Rn is a basis of WQSym * n , 2. (P t ) t∈T Rn is a basis of Prim * n , 3. (P t ) t∈N Rn is a basis of TPrim * n .
Proof. We do a mutually recursive induction on n to prove these three items.
As dim(WQSym * 1 ) = dim(Prim * 1 ) = dim(TPrim * 1 ) = 1 the base case is trivial. By Proposition 1, Item 2 up to degree n implies Item 1 up to degree n. Similarly, Theorem 2 shows that Item 3 up to degree n implies Item 2 up to degree n. By induction it is sufficient to show that Items 1 and 2 up to degree n − 1 implies Item 3 for n.
For all k ∈ N, let π k be the canonical projector on the homogeneous component of degree k of WQSym * . We define π <k : = ∑ k−1 i=0 π i . Fix I = [i 1 , . . . , i p ] with p ≤ n and u a packed word of size n − p. Notice that if p = n we immediately have u = ǫ and ∆ ≺ (Φ I (R ǫ )) = 0. We suppose now that p < n. By Equation (1.26), in the half coproduct ∆ ≺ (Φ I (R u )) all the maximums must be in the left tensor factor, which therefore must be at least of degree i p . By linearity, for all x ∈ WQSym * n−p , . (3.8) Thanks to Corollary 4 for f = [r 1 , . . . , r d ] ∈ F Rn−p , the coproduct∆(P f ) is computed by deconcatenation of forests. So if f ∈ F I Rn−p , in particular r d is of weight at least n − i p + 1 then for j ≥ i p we have n − 1 − j < ω(r d ) so that all the terms in the previous sum vanishe. A similar reasoning applies to ∆ ≻ (Φ I (x)) using the fact that f ∈ F I Rn−p imply 1 ≤ i 1 ≤ ω(r 1 ).
Moreover, by induction we have that {P f | f ∈ F I Rn−p } are linearity independent as {P f | f ∈ F Rn−p } is a basis of WQSym * n−p . Recall the description of N Rn in Remark 70 as a disjointed union of sets depending on I: Since Φ I is injective on WQSym * n−p then {Φ I (P f ) | f ∈ F I Rn−p } are linearly independent. According to Lemma 114, for all f ∈ F I Rn−p we have Φ I (P f ) ∈ Im(τ I ) ∩ TPrim * n . Moreover, thanks to the direct sum of Equation (3.3): and by definition of P, in particular P Node R (I,[], f r ) : = Φ I (P f r ), the family {P t | t ∈ N Rn } are linearly independent. Finally, by cardinalities of Theorem 75 it is a basis of TPrim * n .
Remark 119. The basis P is indexed by red-packed forests (F R ). We will also use redskeletons (F Rske ) or packed words (PW) as index thanks to the bijections of Remark 68 and F R of Definition 63.

Lemma 122.
For any i and α, we have Im(Ψ i α ) = Im(τ i α ) where Im( f ) denotes the image of f . Proof. For any i and α, the inclusion Im(Ψ i α ) ⊂ Im(τ i α ) is automatic by definition of Ψ i α and τ i α and linearity. Indeed, for any w ∈ PW n τ i α (Ψ i α (Q w )) = Ψ i α (Q w ).
More examples can be found in the annexes section with Table 13 and Figure 9. Proof. The proof structure is the same as the one of Theorem 118 except for some statements that are exchanged as we can see in this Lemma 122 Im(Φ I ) = Im(τ I ) and Im(Ψ i α ) = Im(τ i α ).

Isomorphism between WQSym and WQSym *
According to Corollary 3 WQSym (resp. WQSym * ) is freely generated as a dendriform algebra by TPrim (resp. TPrim * ). Therefore, any linear isomorphism between TPrim and TPrim * would lead to a bidendriform isomorphism between WQSym and its dual. Thanks to the two bases P and O any graded bijection between red-irreducible and blueirreducible packed words leads to such an isomorphism. We first make explicit how this is done. Then the bijection is actually obtained as the restriction to red-irreducibles of an involution on all packed words. The definition of the bijection requires a new kind of forest mixing red and blue factorizations, namely bicolored-packed forests.

A combinatorial solution to an algebraic problem
In this Section 4.1, we use the skeleton representation for bases P and O as said in Remarks 119 and 127. Moreover we fix a graded bijection µ between red-irreducible and blue-irreducible packed words.

Definition 128.
Recall that (P t ) t∈N Rn is a basis of TPrim * n (Theorem 118) and (O t ) t∈N Bn is a basis of TPrim n (Theorem 126). By linearity, setting (4.1) for all t = v ∈ N R , defines a linear isomorphism between the vector spaces TPrim * n and TPrim n Definition 129. We define σ µ as the extension of µ from red-skeleton to blue-skeleton forests by: . (4.3) Definition 130. We denote Σ µ the unique bidendriform isomorphism from WQSym * to WQSym which verify for all f ∈ F R : The existence and unicity are guaranteed by Corollary 3.
In the following example, we take µ(212) : = 122 and µ(w) : = w for other words with a size less than 3 as they are simultaneously red-irreducible and blue-irreducible packed words.
Here are all red and blue-irreducible packed words of size 1, 2 and 3: These two forests are the same as those used in Examples 117 and 125. So we have here the first example of the isomorphism from the basis R to the basis Q: We now have a construction of a bidendriform isomorphism for any graded bijection µ between red-irreducible and blue-irreducible packed words.

Full decomposition of packed words into bicolored forests
In order to define a bijection between red-irreducible and blue-irreducible packed words, we need a new kind of forests that mixes up the red and blue factorizations. More precisely, we will recursively alternate these factorizations. We start with an unexpected lemma which implies that starting by red or blue does not matter.
Lemma 133. For all a, b, c ∈ PW, with c = ǫ, the following relations hold: a◮(b c) = b (a◮c) and a◮1 = a 1, (4.5) where 1 is the packed word of size 1.
Proof. Let a ∈ PW and 1 the packed word of size 1, by Definitions 40 and 79 a◮1 = a\1 and a 1 = a\1.
Let a, b, c ∈ PW, with c = c 1 · · · c n of size n > 0. We start by assuming that c n = max(c) which implies that c = φ I (ψ i α (c * )) = ψ i α (φ I (c * )) with I, i, α, c * unique by Lemmas 39 and 78. With this relation we can deduce: The case where c n = max(c) can be decomposed into different particular cases. In each of these cases, it is possible to find a relation with two different writings of c that begin with φ or ψ just like c = φ I (ψ i α (c * )) = ψ i α (φ I (c * )). These cases with the associated relation are: • the case where c is the packed word 1 then c = φ 1 (ǫ) = ψ 1 • (ǫ), • the case where c is of the form c ′ \1 then c = φ 1 (c ′ ) = ψ 1 • (c ′ ), • the more general case where there is more than 1 maximum including the one at the end then c = φ I·c n (c * ) = ψ c • n (φ I (c * )).
In each of these cases it is possible to prove with a similar method that a◮(b c) = b (a◮c).
Example 134 Definition 135. Let w be an irreducible packed word. Let w = x◮u be the red-factorization of w and let u = y z be the blue-factorization of u. Then w = x◮(y z) is called the red-blue-factorization of w. Symmetrically we define w = y ′ (x ′ ◮z ′ ) the blue-redfactorization of w.
Lemma 136. Let w = x◮(y z) be the red-blue-factorization and let w = y ′ (x ′ ◮z ′ ) be the blue-red-factorization of an irreducible packed word w. With these two factorizations, we have that z = z ′ and it is both red-irreducible and blueirreducible packed word. Moreover, • either z = z ′ = 1, y = x ′ = ǫ and x = y ′ • or x = x ′ , y = y ′ . Proof. We start by prooving the case where z = z ′ = 1, y = x ′ = ǫ and x = y ′ . Let w ′ be an irreducible packed word and w = w ′ \1. We have that w = w ′ ◮1 is the redfactorization of w and w = w ′ 1 is the blue-factorization of w. In this case we immediately have that w = w ′ ◮(ǫ 1) is the red-blue-factorization of w and that w = w ′ (ǫ◮1) is the blue-red-factorization of w. Now let w be an irreducible packed word of size n that cannot be written as w ′ \1. In other words, there is a maximum strictly before the last letter of w (∃i < n, w i = max(w)).
We define the two sets of triplet of packed words that verify equations of the factorizations for w: Thanks to Lemma 133 these two sets are equal, we define S(w) : = S RB (w) = S BR (w).
In the red-blue-factorization w = x◮(y z), we maximize the size of x, then we maximize the size of y in the remaining word. In the blue-red-factorization we commute the order of maximizations. We will caracterize S(w) and see the limit of the two maximizations to prove that they can commute.
Let w * be the packed word comming from w where the last letter and all occurences of the maximum are removed and let I = [i 1 , . . . , i p ], i, α such that w = ψ i α (φ I (w * )). By hypothesis, we have that I = ∅. Let w * = w 1 / · · · /w k be the global descent decomposition of w * . Let ℓ be the maximum such that |w 1 / · · · /w ℓ | ≤ i 1 , i 1 being the position of the first maximum of w. Let r be the minimum such that |w r / · · · /w k | ≤ n − i p − 1, i p being the position of the last maximum of w before the last letter. As i 1 ≤ i p by definition, we have that ℓ < r. We can caracterize the set S(w): with r 0 ≤ r and ℓ ≤ ℓ 0 }.
If we define the serie I = ∑ +∞ n=1 i n z n where i n is the number of both red-and-blueirreducible words of size n, then we have the following relation: So far we have seen red-biplane trees and blue-biplane trees. In this section we define red-blue-biplane trees and blue-red-biplane trees, the edges of these trees are of two different colors and the labels are red-and-blue-irreducible packed words. We denote by Node RB (x, f ℓ , f r ) (resp. Node BR (x, f ℓ , f r )) the biplane tree whose edges between the root and the left forest f ℓ are red (resp. blue) and edges between the root and the right forest f r are blue (resp. red).
Definition 138. The bicolored forests F RB (w) and F BR (w) (resp. trees T RB (w) and T BR (w)) associated to a packed word (resp. irreducible packed word) w are defined in a mutual recursive way as follows: • for any packed word w, let w = w 1 /w 2 / . . . /w k be the global descent decomposition, then F RB (w) : = [T RB (w 1 ), T RB (w 2 ), . . . , T RB (w k )], .
• for any packed word w, let w = w 1 /w 2 / . . . /w k be the global descent decomposition, then F BR (w) : .
(notice the same inversion as in Definitions 93 and 98 for F BR .) • for any irreducible packed word w, let w = a◮(b c) be the red-blue-factorization, then T RB (w) : = Node RB (c, F RB (a), F RB (b)). .
• for any irreducible packed word w, let w = b (a◮c) be the blue-red-factorization, then T BR (w) .
Example 139. For this example we write the word w in hexadecimal in order to have a big and clear example, let w = DDDCCCEBBE9FA587653213449. The word w is irreducible so there is only one tree in the forest. To have T RB (w), we start by the bluered-factorization, w = DDDCCCEBBE 9FA 58765321344 9 = 3332224114◮(58765321344 1321).
Then we decompose each sub-word according to their global descents and do blue-redfactorizations recursively until we have only both red-irreducible and blue-irreducible packed words: More examples can be found in the annexes section with Tables 7 to 12.

Definition 140.
There are two types of bicolored trees, the only difference is that colors red and blue are inverted. Let t be a labeled biplane tree. We write t = Node RB (w, f ℓ , f r ) where w ∈ PW, f ℓ = [ℓ 1 , . . . , ℓ g ] is the left forest of t and f r = [r 1 , . . . , r d ] is the right forest of t. We depict t as follows: We say that t is a red-blue-packed tree if it satisfies: or w = 1 is red-irreducible and blue-irreducible, f r and f ℓ are red-blue-packed forest.
Definition 141. The weight of a bicolored-packed tree is the sum of the size of packed words in the nodes.
We have already done it four times (Definitions 61, 71, 96 and 106) and we will do it one last time, to prove that F RB , T RB , F BR , T BR are bijections, we define F * RB , T * RB , F * BR , T * BR and prove that they are the inverse maps.
Definition 142. We define here the maps F * RB , T * RB , F * BR , T * BR that transforms bicoloredpacked forests and trees into packed words. We reverse all instructions of Definition 140 as follows: • for any red-blue-packed forest f = [t 1 , t 2 . . . , t k ], we have .
• for any blue-red-packed tree t = Node BR (c, f ℓ , f r ), we have .
Theorem 143. The maps F RB and F * RB (resp. T RB and T * RB ) are two converse bijections between packed words of size n and red-blue-packed forests (resp. irreducible packed words and red-bluepacked trees) of weight n. That is to say F -1 RB = F * RB and T -1 RB = T * RB . We have the same result with inversions of red and blue.
Proof. It is simple to prove by induction on the size of the trees that domain and codomain are as announced and that the functions are inverse to each other. The proof is similar to the proofs of Lemmas 62 and 97 and Theorems 73 and 108 using Definition 138 of F RB , T RB , F BR , T BR , Definition 140 of bicolored-packed forests and trees and Definition 142 of F * RB , T * RB , F * BR , T * BR . Remark 144. We now have two new families of forests F RB and F BR that are in bijection with packed words and therefore in bijection with red-packed and blue-packed forests. As in Remarks 119 and 127, this gives us two other way to index bases O and P of WQSym and WQSym * .

An involution on packed words
We are now in position to define a bijection between red-irreducible and blue-irreducible packed words. This bijection is actually the restriction of an involution defined on all packed words. Precisely, we will define two transformations on bicolored forests We need to define the notion of mirror transformation of bicolored-packed forests and trees. This transformation is defined from a red-blue to blue-red or from blue-red to red-blue, so in the notations we will use XY instead of RB or BR to point out where the swap is made.
Definition 145. The mirror transformation of a bicolored-packed forest f = [t 1 , . . . , t k ] is given by f : = [ t k , . . . , t 1 ] where t i is the mirror transformation of t i recursively defined as follows. For any t = Node XY (z, f ℓ , f r ) then In other words, it is a recoloration of each edges using the other color. Every blue edges become red and vice versa. More examples can be found in the annexes section with Table 13.
When we focus on the packed words associated to these forest, the color swap correspond to the swap of the two operations ◮ and in a bicolored-factorization. More precisely, if w is an irreducible packed word and w = x◮(y z) is the red-blue-factorization of w, then the color swap on the associated forest correspond to w ′ = x (y◮z).
Lemma 150. Mirror transformation and color swap commute. It means that for all bicoloredpacked forest f , we have f = f . Proof. The proof is immediate. Indeed, the definition of mirror transformation is independant of color swap and symmetrically, the color swap is independant of the tree shape.
Corollary 151. The diagram on Figure 5 is commutative. So w := F -1 RB ( F BR (w)) is an involution on packed words.
Proof. Thanks to Proposition 147 and Lemma 150 the diagram is immediately commutative. The mirror transformation and the color swap are independant involutions so the conjunction is an involution. Figure 5: Commutative diagram of maps on bicolored-packed forests Corollary 152. The application w → w send blue (resp. red) irreducibles packed words to red (resp. blue) irreducibles packed words.
Some examples can be found in th annexes section with Tables 14 and 15.
Proof. If w is a red-irreducible packed word, then the red-blue-factorization of w is of the form w = ǫ◮(y z). Then the color swap correspond to the words w ′ = ǫ (y◮z) which is blue-irreducible.

Main theorem
In Section 4.1 we fixed a graded bijection µ between red-irreducible and blue-irreducible packed words. After that, we extend it to all red-skeleton forests as σ µ . We finished by defining Σ µ as a bidendriform isomorphism from WQSym * to WQSym. Now we can set µ : w → w as a graded bijection. The extension σ µ correspond to the color swap on red-packed forests (i.e. σ µ : f → f ). Finally we have the following theorem: Theorem 153. The linear map Σ : WQSym * → WQSym defined as for all packed forest f , is a bidendriform isomorphism between WQSym * and WQSym.
Proof. This theorem is a direct consequence of Corollaries 3 and 152.

Conclusion
The main contribution of this paper is the combinatorial construction of biplane trees. They are the combinatorial ingredient which completes the algebraic theory of Foissy [17] and allows us to describe the explicit isomorphism. Besides, they are also an innovative combinatorial family and open promising research perspective.

Generalization of the inversion of permutation to packed words
The inherent difficulty of finding an explicit isomorphism between WQSym and its dual lies in the fact that there is no "inversion" operation on packed words. Indeed, in the case of FQSym, the Hopf algebra indexed by permutations, the isomorphism is given by the inversion of permutations. The solution we offer, using biplane trees, is actually not a generalization of FQSym in this sense. Indeed, even though permutations are a subset of packed words, the restriction of our involution on packed words to permutation is not the inversion. In particular, if a permutation σ is both red-irreducible and blueirreducible, its image is itself and not its inverse. This is the case for all packed words which are both red-irreducible and blue-irreducible. Nevertheless, our involution is somehow "compatible" with the inversion in the sense that if we arbitrary decide that the image of σ is σ −1 for all σ such that σ is a red-blue-irreducible permutation, then the rest of construction ensures that the image of σ is σ −1 for all permutations (not necessarily irreducible anymore). But we don't know how to define the inversion on red-blue-irreducible elements which are not permutations, which is why to stick with the identity in all case, including permutations. Stays the open question: is there a generalization of the inversion of permutations on packed words? In other words, one would want an involution on packed words which restricts to the inversion on permutations and gives a bidendriform isomorphism between WQSym and its dual. A consequence of our work is that it is sufficient to find such an involution on red-blue-irreducible packed words.

Generalization of the biplane trees to parking functions
A long term goal would be to somehow generalize the structure of biplane trees to all bidendriform Hopf algebra. The first step would be to look at the Hopf algebra indexed by parking functions PQSym. Indeed PQSym is also a bidendriform bialgebra and parking functions are a superset of the packed words. The question of generalizing the structure to parking functions involves both combinatorics and algebra. The first thing is to compute bases of PQSym in which the shuffle product is not shifted. It can be done with a generalization of Definition 29 [10].
The lines of research induced by this work are the following: • How to generalize biplane tree structure to PQSym? • We will then look for what are the necessary and sufficient ingredients to develop biplane tree structures and obtain bidendriform automorphisms on all bidendriform bialgebras.

Link between bidendriform bialgebras and skew-duplicial operad
As said in Remarks 48 and 88 the operations ◮ and / (resp. and /) unexpectedly verify relations of the skew-duplicial operad [23]. These relations reveal a new application of the skew-duplical operad applied on packed words.
• Can we find a skew-duplical structure on WQSym which is linked to the bidendriform structure? • More generally, is there a link between bidendriform bialgebra and skew-duplical?

Link between bidendriform bialgebras and L-algebras
As said in Remark 56, the sequence that count unlabeled biplane trees is the dimensions of the free L-algebra on one generator (see [25]). It would be interesting to investigate the link between L-algebra and bidendriform bialgebras through the use of biplane trees.
The study of the operad on the three operations {◮, , /} is a start in order to study the link between bidendriform bialgebras and the skew-duplical operade or L-algebras.

Annexes
In Tables 7 to 12 we have red-packed forests, blue-packed forest and bicolored-packed forests associated to all packed words of size smaller than 4. In Table 13 we have the isomorphism between WQSym (bases O and Q) and its dual (bases P and R) for size smaller than 3. Basis O and P are indexed by bicolored-packed forests. This illustrates the main theorem of Section 4. 4. In Table 14 we have the involution of Corollary 152 for all packed words of size 4. They are organized by evaluations. Red-irreducible (resp. bleu-irreducible) packed words are underlined in red (resp blue) in the first (resp. second) column.
In Table 15 we have the involution of Corollary 152 for all red-irreducible packed words that are not blue-irreducible. It correspond to words underlined in red in front of a word underlined in blue in Table 14.