The product of trees in the Loday-Ronco algebra through Catalan alternative tableaux
aa r X i v : . [ m a t h . C O ] D ec THE PRODUCT OF TREES IN THE LODAY-RONCO ALGEBRATHROUGH CATALAN ALTERNATIVE TABLEAUX
J.-C. AVAL, X. VIENNOTThe aim of this note is to show how the introduction of certain tableaux, called
Catalan alternative tableaux , provides a very simple and elegant description of theproduct in the Hopf algebra of binary trees defined by Loday and Ronco. Moreover,we use this description to introduce a new associative product on the space of binarytrees. Introduction
Loday and Ronco defined in [5] an interesting Hopf algebra structure on the linearspan of rooted planar binary trees. This algebra is defined as a sub-algebra of theMalvenuto-Reutenauer Hopf algebra of permutations. Let S n be the symmetric groupand k be a ground field. We denote by k [ S n ] the group algebra. Malvenuto andReutenauer construct in [6] a Hopf algebra structure on k [ S ∞ ] = M n ≥ k [ S n ] . It is worth to recall here that the Malvenuto-Reutenauer algebra contains the sumof Solomon descent algebras
Sol ∞ = L n ≥ Sol n with Sol n of dimension 2 n − .In [5], Loday and Ronco define a sub-Hopf algebra of k [ S ∞ ]: k [ Y ∞ ] = M n ≥ k [ Y n ]where Y n is the set of planar binary trees with n internal vertices.The aim of this work is to present a very simple presentation for the product oftwo trees in k [ Y ∞ ] through the use of Catalan alternative tableaux . These objectswere introduced by X. Viennot [12] as a special case of alternative tableaux, whichare in bijection with permutation tableaux. Permutations tableaux were introducedby E. Steingrimsson and L. Williams [9], as a subclass of Γ-diagram defined by A.Postnikov [8]. This notion was used by S. Corteel and L. Williams [2, 3] in the studyof the physical model named PASEP (partially asymmetric exclusion process), seefor example the seminal paper by B. Derrida and al. [4]. These tableaux are alsorelated to the study of total positivity for Grassmannian [13]. Both permutation andalternative tableaux are in bijection with permutations, see for example P. Nadeau[7]. The advantage of alternative tableaux is to preserve the symmetry between rowsand columns.
This research has been supported by the ANR (project MARS/06-BLAN-0193).
This new interpretation of the Loday-Ronco product motivates the introduction ofa new associative product, that we call the
The Loday-Ronco Hopf algebra
We recall the definition of the Loday-Ronco product of binary trees. Since thisproduct is inherited from the Malvenuto-Reutenauer product of permutations, weshall first recall the definition of the product in k [ S ∞ ], denoted by ∗ . We refer to [6]for more details and only recall briefly the definition.Let u = u u . . . , u k be a k -tuple of distinct integers. We define the standardization of u and denote it by Std( u ) as the unique permutation σ ∈ S k that preserves therelative order of the u i ’s, i.e. σ i < σ j ⇐⇒ u i < u j . For example, Std(3275) = 2143. Conversely, for σ ∈ S k a permutation and A = { a , a , . . . , a k } a set of k (distinct) integers, we define σ | A the k -tuple with distinctentries in A such that Std( σ | A ) = σ . With this notation we may define the product ∗ in k [ S ∞ ] as follows. Let σ ∈ S k and τ ∈ S l . We set σ ∗ τ = X A ⊔ B = { , ,...,k + l } σ | A .τ | B where ⊔ denotes the disjoint union, and . stands for concatenation.For example :12 ∗
213 = 12 435+13 425+14 325+15 324+23 415+24 315+25 314+34 215+35 214+45 213 . Remark 1.
The product ∗ that we consider is sometimes known as the product inthe dual Malvenuto-Reutenauer algebra. But it is the one used in [5] to define theLoday-Ronco algebra, that we shall now describe.
Let Y n denote the set of binary trees with n internal vertices. We recall that thecardinality of Y n is given by the n -th Catalan number C n = n +1 (cid:0) nn (cid:1) .Let ˜ Y n denote the set of increasing binary trees , i.e. of binary trees such that eachinternal vertex has a distinct label in { , . . . , n } , and such that the labels increasealong the tree.It is well known that increasing binary trees are in bijection with permutations: toobtain the permutation from the tree, you just have to read the labels from left toright.Below is an example of a plane binary tree with 8 internal vertices, with a increasingbinary tree with the same underlying tree, and with the corresponding permutation σ ∈ S n . RODUCT OF TREES AND CATALAN ALTERNATIVE TABLEAUX 3
12 3 46 7 85 σ =
We denote by Ψ : S n → Y n the composition of the bijection S n ≃ ˜ Y n with theprojection ˜ Y n → Y n which consists in forgetting the labels. The induced linear mapΨ : k [ S n ] → k [ Y n ] has a linear dual Ψ ∗ : k [ Y n ] → k [ S n ] obtained by identifying eachbasis with its own dual. For exampleΨ ∗ (cid:16) (cid:17) = 3412 + 4312 + 2413 + 4213 + 2314 + 3214 . We also define for any tree T the set Z T = { σ ∈ S n / Ψ( σ ) = T } so that Ψ ∗ ( T ) = P σ ∈ Z T σ .The inclusion map Ψ ∗ gives rise to a graded linear map Ψ ∗ : k [ Y ∞ ] → k [ S ∞ ] andthe main result in the construction of the Loday-Ronco algebra may now be statedas (Theorem 3.1 in [5]): Theorem 2.
The image of the inclusion map Ψ ∗ : k [ Y ∞ ] → k [ S ∞ ] is a sub-Hopfalgebra of k [ S ∞ ] . So, k [ Y ∞ ] inherits a structure of Hopf algebra. Trees and Catalan alternative tableaux
We now present the Catalan alternative tableaux. Let us denote by N the set ofnonnegative integers. A Catalan alternative tableau in given by • a path in N × N from { } × N to N × { } made of (0 ,
1) and (1 ,
0) steps. Thelength of the path is called the size of the tableau, and the cells below the pathare simply called the cells of the tableau . The path defining the tableau can becalled the shape of the tableau. • a set of blue and red dots in the cells of the tableau such that:(1) there is no dot below a red dot;(2) there is no dot on the left of a blue dot;(3) any cell of the tableau is either below a red dot, or on the left of a bluedot.Let us give an example of a Catalan alternative tableau of size 23. J.-C. AVAL, X. VIENNOT (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
It is possible to directly check that Catalan alternative tableaux are enumeratedby Catalan numbers (whence their name), but we shall use the following proposition,more adapted to our context.
Proposition 1.
The Catalan alternative tableaux of size n − are in bijection withbinary trees with n internal nodes.Proof. We refer to [10] (algorithm 2.2) for a formal proof and give here only theidea of the construction. In fact, in that paper, the algorithm was given in term of“Catalan permutation tableaux”, the subclass of permutation tableaux correspondingto Catalan alternative tableaux, and discussed in [9].We start with Catalan alternative tableau of size n − shift , and two cases are to be distinguished: • if the corresponding corner in the tableau is blue: ts us t u and we erase the row of the corner in the tableau; • if the corresponding corner in the tableau is red: s u ts t u and we erase the column of the corner in the tableau.On the example, we obtain: RODUCT OF TREES AND CATALAN ALTERNATIVE TABLEAUX 5 = It is not difficult to verify that this construction is a bijection. (cid:3)
For a permutation σ ∈ S n , its Up-Down sequence ( cf. [11]) is the vector Q ( σ ) =( q , . . . , q n − ) ∈ {− , +1 } n − such that q i = +1 iff σ i +1 > σ i . It is clear that for any tree T , all the σ in Z T have the same Up-Down sequence,which we may call the Up-Down sequence of T , also called canopy of the binary tree T in [11].Now we may view the shape of a Catalan alternative tableau of size n − {− , +1 } n − (horizontal steps correspond to “-1” entries and vertical stepsto +1 entries).We have the following property: Proposition 2.
The shape of the tableau associated to a tree T through the bijectiondescribed in Proposition 1 is the Up-Down sequence of T , as well as the commonUp-Down sequence of any permutation σ in Z T . Now the algorithm described above may be extended to labelled trees: we may put n labels on the shape of a Catalan alternative tableau of size n − If we keep the labels of the nodes when we apply the algorithm to get a tree fromthe tableau, we obtain a labelled tree
97 2 1 5 846 3
In the previous example, we may say that the tableau was labelled with the permu-tation 792531648. As a consequence of the bijection, we get the following property.
J.-C. AVAL, X. VIENNOT
Proposition 3.
Let σ be a permutation of size n . We may label a tableau C with σ , then apply the bijective algorithm. The labelled tree that we obtain is increasing ifand only if: • the shape of C is the Up-Down sequence of σ ; • the position of the red and blue dots in C is the only one which gives the binarytree Ψ( σ ) . The product of trees through Catalan alternative tableaux
Now we come to the main result of this work.
Theorem 3.
Let T and T be two binary trees. Their product in the Loday-Roncoalgebra T ∗ T = X T is given by taking the sum over the trees T associated to Catalan alternative tableauxin the union U C C ? CC ? where C and C are the Catalan alternative tableaux associated respectively to T and T , and the question mark ( ? ) represent any (valid) placement of (red and blue)dots in the rectangles.Proof. By definition of Ψ ∗ , we have:(1) Ψ ∗ ( T ∗ T ) = Ψ ∗ ( T ) ∗ Ψ ∗ ( T ) = X σ ∈ Z T ∗ X σ ∈ Z T σ = X σ ∈S σ. Let σ be an element of S . By definition of the product ∗ in the Malvenuto-Reutenauer algebra, σ is of the form: σ = τ .τ (concatenation) with the lettersappearing in τ and τ form a partition of { , . . . , n } , and(2) Ψ( τ ) = T and Ψ( τ ) = T . Thus if Ψ( σ ) = T ,the Up-Down sequence of T Q ( T ) is either Q ( T ) U pQ ( T ) or Q ( T ) DownQ ( T ). Hence the form of the Catalan alternative tableau C associatedto T is one of the two given in the Theorem 3. We label the shape of C with theentries of σ . The red and blue dots in C have to be placed in a position such thatby applying the bijective algorithm, we obtain an increasing binary tree. But if weapply the algorithm to the part of C that carries the entries of τ (respectively τ ), RODUCT OF TREES AND CATALAN ALTERNATIVE TABLEAUX 7
Propositions 1 and 3 imply that the (red and blue) dots of C in the correspondingsubparts of C have to be placed in the same configuration than in C (respectively C ). Thie implies that C has the required form.Conversely, let T be a tableau of the form described in the Theorem 3, and σ ∈ Z T .By cutting σ in two parts u and u of lengths the sizes of C and C , we may write: σ = u .u with Std( u ) = τ and Std( u ) = τ .It is again a simple application of Propositions 1 and 3 that we have: Ψ( τ ) = T and Ψ( τ ) = T , which was to be proved to complete the proof of Theorem 3. (cid:3) The product of binary trees
In light of Theorem 3, it seems natural to introduce a new product on k [ Y ∞ ] asfollows. Definition 4.
We define the product of two binary trees T and T , associatedrespectively to Catalan alternative tableaux C and C by: T T = X T where the sum is taken over the trees T associated to Catalan alternative tableaux inthe set: C ? C and the question mark ( ? ) represent any (valid) placement of (red and blue) dotsin the rectangles. It is clear that this defines an associative product on k [ Y ∞ ]. It is worth to notethat for T ∈ Y k and T ∈ Y l , then the product T T is in Y k + l − (in this case thenumber of internal edges is preserved).We give below an example of this product, that should be checked by the reader. J.-C. AVAL, X. VIENNOT
Acknowledgement.
The authors sincerely thank F. Chapoton for fruitful discus-sions.
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LaBRI, Universit´e Bordeaux 1, 351 cours de la Lib´eration,33405 Talence cedex, FRANCE
E-mail address : [email protected] URL : (Xavier Viennot) LaBRI, Universit´e Bordeaux 1, 351 cours de la Lib´eration, 33405Talence cedex, FRANCE
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