Hopf algebra of permutation pattern functions

Transcription

1 Hopf algebra of permutation pattern functions Yannic Vargas To cite this version: Yannic Vargas. Hopf algebra of permutation pattern functions. Louis J. Billera and Isabella Novi. 26th International Conference on Formal Poer Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Poer Series and Algebraic Combinatorics (FPSAC 2014), pp , 2014, DMTCS Proceedings. <hal > HAL Id: hal Submitted on 1 Oct 2015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, hether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 FPSAC 2014, Chicago, USA DMTCS proc. AT, 2014, Hopf algebra of permutation pattern functions (Extended abstract) Yannic Vargas Laboratoire de Combinatoire et d Informatique Mathématique (LaCIM), UQAM, Canada Abstract. We study permutation patterns from an algebraic combinatorics point of vie. Using analogues of the classical shuffle and infiltration products for ord, e define to ne Hopf algebras of permutations related to the notion of permutation pattern. We sho several remarable properties of permutation patterns functions, as ell their occurrence in other domains. Résumé. On étudie les motifs de permutations d un point de vue de la combinatoire algébrique. En utilisant des analogues des produits classiques de mélanges et d infiltration, on définit deux nouvelles algèbres de Hopf de permutations reliées à la notion de motifs de permutations. On montre quelques identités remarquables des fonctions de motifs de permutations, ainsi que ses liens avec d autres domaines. Keyords: Pattern permutation, shuffle, infiltration, bialgebra, Hopf algebra, representative function 1 Introduction Let N the set of nonnegative integers. For n N, consider S n the set of permutations on {1, 2,..., n}. We denote λ the empty permutation, thus S 0 = {λ}. Also, let S = n 0 S n (disjoint union) the set of all permutations. Let Q S be the set of all functions from S to Q. This is a commutative Q-algebra hen it is equipped by sum and the pointise product: if f, g Q S and σ S, then (f + g)(σ) = f(σ) + g(σ), (f g)(σ) := f(σ)g(σ). In this or e consider a special subalgebra of Q S based on permutation patterns. If, σ S, then σ is said to be a pattern of if there exists a subsequence of entries of that has the same relative order as σ. We define { σ} as the number of occurrences of σ as a pattern of. For example, { } = {254, 253, 143} = Notice that { λ} = 1 and { } = 1, for all S. Permutation patterns are considered in several domains, such as as theoretical computer science and enumerative combinatorics; namely, in problems lie stacsorting (Knuth, 1968 [8]) and pattern avoidance. This or is intended to study permutation patterns from vargas lozada.yannic@uqam.ca. Partially supported by ISM and LaCIM c 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

3 840 Yannic Vargas an algebraic combinatorics point of vie. The main theorem of this article (proved in section 5) is the folloing: Theorem 1 The family of functions { σ}, for σ S, span a free commutative subalgebra of Q S. To describe the algebraic structure in the above theorem, e introduce to ne products in the vector space of permutations QS: the super-shuffle, and the super-infiltration (see sections 3.2 and 3.3, respectively). The first one interpolates beteen to products introduced by Malvenuto and Reutenauer in [10] (section 3, page 978). The super-infiltration is a natural generalization of the classical infiltration product for ords (see [9], chapter 6.3, page 128). We also consider the folloing coproduct in QS: (σ) = α β, σ = α β here is defined as follo: if α = a 1 a n S n and β = b 1 b p S p, then α β := a 1 a n (b 1 + n) (b p + n) S n+p. Theorem 2 The Q-vector space QS, endoed ith the product of super-infiltration and the coproduct, is a Hopf algebra. As an algebra, it is a free commutative Q-algebra. The function ( { }) σ σ is an isomorphism from this algebra to the subalgebra of theorem 1. Theorem (2) (proved in section 4) clearly implies Theorem (1). In particular, e have an analogue of a relation due to Chen, Fox and Lyndon (see [2]; also [9], page 131). This relation describes the algebraic structure of the subalgebras of theorems 1 and 2: { }{ } α β = (α β, σ) σ S { }. (1) σ We finish this or by shoing several identities and properties of permutation pattern functions, as ell some occurrence of this objects in other domains. These results are expanded on and proven in the manuscript [16] of the same name. 2 Classical binomial coefficients of ords, shuffle and infiltration In this section e present some basic notions on ords and binomial coefficients of ords. Most of the proofs of the results mentioned here can be found in [9]. The diligent reader ill notice that these facts are straighforard to verify. 2.1 Operations on ords Let N be the set of non-negatives integers. For all n N e rite [n] := {1, 2,..., n} and [0] :=. An alphabet is a set A. The letter of the alphabet are the elements of A. A ord over the alphabet A is a finite sequence of elements of A. The set of all ords over A is denoted by A. The set A is a monoid ith the concatenation product and the empty ord, denoted by λ, as neutral element for the concatenation.

4 Hopf algebra of permutation pattern functions 841 Let = a 1 a 2 a n be a ord in A. The length of, denoted by, is the number n of letters of. If I = {i 1 < i 2 < < i } is a subset of [n], let I := a i1 a i2 a i. A ord u A is said to be a subord of if there exist a subset I of [n] such that u = I. We rite u if u is a subord of. If = n and I [n] is fixed, e say that [n]\i is the complementary subord of I in. The alphabet of a ord A is the set alph() of the letters ocurring in. If the alphabet A is totally ordered, and u, v A are such that all letters in v are greater than the letters in u, e rite alph(u) < alph(v). 2.2 Formal series and non-commutative polynomials Let A be a countable ordered alphabet and K a closed field. A formal series ith coefficients in K and variables in A is an infinite formal linear combination f = (f, ), A here (f, ) is the value of f on. It is called the coefficient of in f. The set of all series is denoted by K A. It acquires a K-algebra structure, ith (f + g, ) = (f, ) + (g, ) and (fg, ) = (f, u)(g, v). =uv The space K A is endoed ith the folloing bilinear form: for all f, g K A, f, g := (f, )(g, ). A 2.3 Shuffle and infiltration products If u A n and v A p, e define the shuffle product and the infiltration product of u and v as u v :=, u v :=. (2) Example 2.1 I J = [n+p] I = u, J = v I J [n+p] I = u, J = v ba ab = baab + baab + baba + abab + abba + abba = 2 abba + abab + 2 baab + baba. ba ab = bab + aba + baab + baab + baba + abab + abba + abba = bab + aba + ba ab = aba + bab + 2 abba + abab + 2 baab + baba. We remar that the infiltration product contains the shuffle product. The shuffle product as introduced by Eilenberg and MacLane in [3]. Chen, Fox and Lyndon introduced in [2] (Theorem 3.9, page 93) the infiltration product in an algebraic frameor; Ochsenschläger (see [12]) defined this product independently in the context of binomial coefficients of ords (see the next section). Both products are associative and commutative (see [9], proposition and proposition ).

5 842 Yannic Vargas The shuffle and infiltration product are related by the folloing identity ([9], Theorem ): (u A ) (v A ) = (u v) A, (3) here A is the formal series A := A and denotes the Hadamard product. For f, g K A and A, it is defined by (f g, ) := (f, )(g, ). Both products can also be defined from a general recursive formula. Let Ā := A {0}, here 0 is the zero of K. A quasi-shuffle product in K A is a bilinear product such that λ = λ = for any ord, and, for any a, b A and u, v A, here [, ] : Ā Ā Ā is a function satisfying: (S1) [a, 0] = 0, for all a Ā; (S2) [a, b] = [b, a], for all a, b Ā; (S3) [[a, b], c] = [a, [b, c]], for all a, b, c Ā; au bv = a(u bv) + b(au v) + [a, b](u v), (4) This construction is old and has been rediscovered many times. As mentioned by one the referees, it probably appeared first in a or of Neman and Radford from 1979 [11]. In the last decade, it appeared in or of Hazeinel and (in a particular case) in or of Hoffman. Enjalbert and Minh consider the definition above ([4], definition 2.1). Hoffman defined this product ith an additional graded property on the ords in A (see [6], page 51). When [a, b] := 0 for all letters in A \ {λ}, e obtain the shuffle product; if [a, b] := δ a,b a, e obtain the infiltration product. Consider no the deconcatenation coproduct: () = u v. (5) Proposition 2.2 (K A,, ) is a bialgebra. =uv See [4], proposition 2.3, for a proof and [6], theorem 3.1, for the graded case. 2.4 Binomial coefficients of ords An important tool for counting subords is the notion of binomial coefficient of ords, a generalization of the classical binomial coefficient for integers. Given u, v A, the number of subords of u that are equal to v is called the binomial coefficient of u by v and is denoted by ( u v). For example, ( ) bbab = 2. bab If m, n N are such that n m and a is a letter in A, e remar that ( ) ( ) a m m a n =, n here the right hand side denotes the classical binomial coefficient. of integers.

6 Hopf algebra of permutation pattern functions 843 The binomial coefficients of ords are closely related to the shuffle and infiltration products. For instance, the coefficient of u in the infiltration product of u and v is ( ) u (u v, u) = {(I, J) : I J [ u + v ], u I = u, u J = v} = {J [ u ] : u J = v} =. v A ay to encode all subords of a fixed ord is given by the Magnus transformation: it is the algebra endomorphism µ of Z A defined by µ(a) = λ + a, for all a A. For example, Is easy to see that µ(bbab) = (λ + b)(λ + b)(λ + a)(λ + b) = λ + a + 3 b + ab + 2 ba + 3 bb + 2 bab + bba + bbb + bbab. µ(u) = v A ( ) u v, (6) v for all u A. The Magnus transformation is an automorphism ([9], cor ). The inverse is given by µ 1 (u) = v A ( 1) u + v ( ) u v. (7) v It can be shon (by induction on the length) that the shuffle of a ord v A ith the formal series A is: v A = ( ) u u (8) v u A (see [9], prop ). This implies that shuffling ith A is the adjoint operator of the Magnus transformation: ( ) u µ(u), v = = v A, u. (9) v From the series relation (3) involving shuffle and infiltration, e obtain for all u, v, A. By (9), this is equivalent to u A, v A, = (u v) A,, u, µ() v, µ() = u v, µ(). This implies the folloing product rule beteen binomial coefficients of ords ( )( ) u v = ( (u v, s) s s A ). (10) Formally, the equation above means that the span of the set of functions { ( u) : u A } is a subalgebra of the algebra of functions from A to Q, here the ring structure is defined pointise. In the next sections e develop an analogue of this result for permutation pattern functions.

7 844 Yannic Vargas 3 Permutation pattern, supershuffle and superinfiltration 3.1 Operations on permutations Let n N. A permutation on [n] is a bijection from [n] to [n]. Consider S n the set of permutations on [n]. We denote λ the empty permutation, thus S 0 = {λ}. Also, let S = n 0 S n (disjoint union) the set of all permutations. Let A be a total ordered alphabet. The standard permutation of a ord = a 1 a n on A is the unique permutation, denoted st(), such that for any i, j [n], st() i < st() j a i < a j or (a i = a j and i < j). The permutation st() may be obtained by numbering from left to right the letters of, starting ith the smallest letter, then the second smallest letter, and so on. For example, if a < b < c e have If σ S n and N let u c b b a b b a c b st(u) [σ] := (σ 1 + )(σ 2 + ) (σ n + ). The set of permutations S is an associative monoid ith the product α β := α[β] n = α 1 α n (β 1 + n) (β p + n) S n+p, (11) if α = α 1 α n S n and β = β 1 β p S p. A permutation σ is said indecomposable if σ = α β implies α = λ or β = λ. Let Ind the set of the indecomposable permutations. Then, S is a free monoid generated by Ind. 3.2 Permutation pattern If, σ S, then σ is said to be a pattern of if there exists a subsequence of entries of that has the same relative order as σ. We define { σ} as the number of permutation patterns of σ in. In terms of standardization, this number can be described as { } = { u : st(u) = σ}. (12) σ Notice that { { λ} = 1 and } = 1, for all S. Example 3.1 { } = {514, 513} = Some combinatorial numbers can be deduced from the number of permutation patterns of a permutation. For instance, if n, e have Also, if σ S, then { } 12 n = 12 { } σ = coinv(σ) and 12 ( ) n. (13) { } σ = inv(σ), (14) 21 here coinv(σ) and inv(σ) are the number of coinversions and inversions of σ, respectively: coinv(σ) := {(i, j) : i < j and σ i < σ j }, inv(σ) := {(i, j) : i < j and σ i > σ j }. (15)

8 Hopf algebra of permutation pattern functions Supershuffle In [10], Malvenuto and Reutenauer introduced to products in KS. If α S n and β S p, let α β := =uv st(u)=α, st(v)=β alph(u) alph(v)=[n+p] and α β := α [β] n = I J=[n+p] I =α, J =[β] n alph( I ) alph( J )=[n+p] It is proved that (KS, ) and (KS, ) are isomorphic algebras. The isomorphism is given by θ : KS KS, here θ is defined on permutations by θ(σ) = σ 1. It is possible to rerite these to products using the notion of binomial coefficients of ords: if u, v, are ords on an alphabet A, let ( u,v) be the number of complementary subords (1, 2 ) of such that 1 = u and 2 = v. Using the isomorphism θ, e have α β = S n+p ( ) α 1, [β 1 1 and α β = ] n S n+p ( ). α, [β] (16) n We define a ne product in KS hich interpolates beteen and. If α S n and β S p, the supershuffle of α and β is α β :=, (17) I J=[n+p] st( I )=α, st( J )=β alph( I ) alph( J )=[n+p] The difference beteen the definitions of α β, α β and α β is indicated in blue belo. If S n+p, let { α,β} := {(I, J) : I J = [n + p], st(i ) = α, st( J ) = β}. Then, by definition of e have α β = S n+p { }. (18) Observe that { } { α,β = } { β,α. So, the product is commutative. Also, e remar that { α,λ} = α}, for all, α S. It is straightforard to prove that is associative (see [16] for more details). Example = ( ) = ( ) + 3 ( ). We have, for example, (21 12, 2341) = { ,12} = {(21, 34), (31, 24), (41, 23)} = 3.

9 846 Yannic Vargas 3.4 Superinfiltration We present no an analogue of the infiltration product for permutations. Let α S n, β S p. The superinfiltration of α and β is α β :=. (19) I J [n+p] I =n, J =p st( I )=α, st( J )=β Let [ α,β] := (α β, ). The superinfiltration contains the supershuffle, as the infiltration contains the shuffle product of ords. Indeed: [ ] = and { } = u (u, v) : { (u, v) : u, v st(u) = α, st(v) = β, v st(u) = α, st(v) = β v complementary subord of u in } (20). (21) From (20), it is clear that the superinfiltration is commutative. Also, it is straightforard to prove that is associative (see [16] for more details). As for the supershuffle product, e remar that [ { α,λ] = α}. Example 3.3 We have: [ ] 2341 = {(231, 21), (231, 31), (231, 41), (241, 21), 231, 21 Then: (241, 31), (241, 41), (341, 21)(341, 31), (341, 41)} = 9. [ ] 231 = {(231, 21), (231, 31)} = , = (22) The map θ behaves ell ith the supershuffle and the superinfiltration product. Proposition 3.4 Let, S. Then { } { α,β = 1 } [ α 1,β and ] [ 1 α,β = 1 ] α 1,β. In particular, 1 θ(α β) = θ(α) θ(β) and θ(α β) = θ(α) θ(β). 4 Hopf algebras of supershuffle and superinfiltration For basic definitions and facts about bialgebras and Hopf algebras see [5], [7] or [15]. The supershuffle and superinfiltration products satisfy the folloing property: Lemma 4.1 Let σ, π, S. Then { } σ π = { σ α, β α=α α β=β β }{ π α, β } and [ ] σ π = [ ][ σ α, β α=α α β=β β π α, β ]. (23)

10 Hopf algebra of permutation pattern functions 847 These identities can be seen in an algebraic frameor (i) : let µ and µ be the bilinear extensions of the supershuffle and superinfiltration products, respectively, and consider the map : KS KS KS defined by (σ) := α β. (24) σ=α β Then, the lemma above said that is a homomorphism for both products µ and µ. In analogy ith proposition (2.2), e have: Theorem 4.2 (KS, µ, ) and (KS, µ, ) are bialgebras. By a theorem of Milnor and Moore, every graded and locally finite bialgebra possesses an antipode, and then it is a Hopf algebra (for more details see for example [1], page 18, section 5). Since (KS, µ, ) satisfies these hypothesis, e have Theorem 4.3 (KS, µ, ) is a Hopf algebra. The algebra (KS, µ ) is not graded by the length (see the example (3.3)) and then an antipode does not come for free from the the theorem of Milnor-Moore. Hoever, the bialgebra (KS, µ, ) indeed possesses a structure of Hopf algebra. The description of the antipodes of (KS, µ, ) and (KS, µ, ) is discussed in the section (4.2). The application is a variant of the coproduct in the Malvenuto-Reutenauer Hopf algebra, ith respect to its monomial functions introduced by Aguiar and Sottile [1]. We remar that is not cocommutative. 4.1 (KS, µ, ) and (KS, µ, ) are free and cofree. Let A a totally ordered alphabet and < lex the lexicographic order on A. Recall that a Lyndon ord over A is a non-empty ord hich is strictly smaller than each of its proper right factors, for < lex. We denote the set of all Lyndon ords on A by L. Any ord A can be factorized in a unique ay as a decreasing product (for lexicographic order) of Lyndon ords (see [2], [9] or [15]). In other ords, = l r1 1 lr, here l i L and l i > lex l j if i < j. We ill use the folloing technical result: Lemma 4.4 (Radford[14]) Let = l r1 1 lr A, ith l 1,..., l L. The polynomial Q = 1/(r 1! r!) l r1 1 l r can be ritten as + R, here R contains only ords of eight equal to the eight of, but smaller than (for the lexicographic order). In order to prove Theorem 2, e ill need to introduce analogous notions for permutations. First, consider the folloing total order on the set of indecomposable permutations Ind: if Ind, e define α < β if α < β, or α = β and α < lex β. Then, the elements in Ind are totally ordered as 1 < 21 < 231 < 312 < 321 < 2341 <. We define a Lyndon permutation as a permutation σ such that σ λ, and for each factorization σ = α β, ith λ, one has σ < β. We denote the set of all Lyndon permutations by L S. (i) These identities are also related ith the notion of section coefficients of a bialgebra, as in [7]. We discuss this notion related to permutation pattern function in a future or.

11 848 Yannic Vargas Lemma 4.5 Any permutation σ Ind S can be factorized in a unique ay as a decreasing product (for the order <) of Lyndon permutations. Consider no the folloing variation of the shuffle product: let be the relative shuffle to the alphabet Ind. Formally, if σ, π Ind S and Ind, and σ λ = λ σ = σ. For instance, (α σ) (β π) := α (σ (β π)) + β ((α σ) π), (25) (21 1) 21 = = ( ) = (21435). For each σ S, let σ be the relative length of σ over Ind; it is the number of letters of σ as a ord in the alphabet Ind. For example, = 3, because = Lemma 4.6 Let α S n, β S p. Then α β = α β + sum of permutations π S n+p such that π < α + β. (26) Proof of theorem (2): Let l 1,..., l L S, ith l i S ni and l i = s i. Consider σ = l r1 1 l r S n, for some n N. By the lemmas (4.4) and (4.6) e have l r1 1 l r = r 1! r! (σ + sum of permutations π S n such that π < σ) + sum of permutations π S n such that π < r 1 s r s l r1 1 l r = r 1! r! (σ + sum of permutations π S n such that π < σ) + sum of permutations π S n such that π < r 1 s r s + sum of permutations π S p, p < n. By triangularity, (KS, ) and (KS, ) are freely generated by the elements in L S. Also, by the dual version of the theorem 6.1 in [13], (KS, ) is cofreely generated by the elements in Ind. Then, (KS, µ, ) and (KS, µ, ) are free and cofree bialgebras. 4.2 The antipodes of (KS, µ, ) and (KS, µ, ) We give no a description of the antipode for (KS, µ, ) and (KS, µ, ). The proof follos easily by induction on the relative length of a permutation. Proposition 4.7 Let σ = σ 1 σ 2 σ n a permutation such that σ = n. Then (KS, µ, ) and (KS, µ, ) possess antipodes S and S, respectively, given by S (σ) = ( 1) (σ 1 σ i1 ) (σ i1+1 σ i1+i 2 ) (σ i1+ +i 1 +1 σ n ), (i 1,...,i ) Comp n S (σ) = ( 1) (σ 1 σ i1 ) (σ i1+1 σ i1+i 2 ) (σ i1+ +i 1 +1 σ n ), (i 1,...,i ) Comp n here Comp n is the set of compositions of n. Corollary 4.8 (KS, µ, ) is a Hopf algebra.

12 Hopf algebra of permutation pattern functions Hopf algebra of permutation pattern functions For each σ S, let ϕ σ : S Q be defined by ϕ σ () := { }, (27) σ for all S. We call ϕ σ the permutation pattern function related to σ. It is straightforard to verify that the collection {ϕ σ : σ S} is linearly independent in Q S. Proposition 5.1 Let S. The pointise product ϕ σ ϕ β in Q S is given by ϕ α ϕ β = [ ] σ ϕ σ. (28) σ S This implies the formula (1). Let PPF = Q{ϕ σ : σ S}. We equip the vector space PPF ith the coproduct (ϕ σ ) := ϕ α ϕ β. (29) σ=α β Theorem 5.2 The vector space PPF, ith the pointise product and the coproduct, is a free and cofree Hopf algebra, isomorphic to (QS, µ, ) by the map σ ϕ σ. It is a subalgebra of Q S, freely generated by the set {ϕ σ : σ L S }. A characterization of the permutation pattern functions permutation pattern functions, as follo: Theorem 3 Let s : S N be a nonzero function such that, for every S, s(α)s(β) = [ ] σ s(σ). σ S Then, there exists σ S such that s = ϕ σ. The relation (28) permit to describe the Magnus transformation for permutation patterns. Several identities for the binomial coefficients of ords can be transfered to the frameor of permutation pattern functions. We have the folloing analogue of the Magnus transformation (compare ith (6) and (7)): let M := σ S ϕ σ σ. Then: Theorem 4 In the monoid-algebra Z[S] of the free monoid (S, ), the function M() define an automorphism. The inverse of this map is given by M 1 () = { } ( 1) + σ σ. σ σ S Permutation pattern functions as representative functions The space Q S is a (S, )-module via (f σ)(π) := f(σ π), for all f Q S and σ, π S. Recall that a function f Q S is representative if the set {f σ} σ S span a finite dimensional Q-vector space. Let α, σ S. If e put β = λ in the first formula of (4.1), e obtain { } σ π (ϕ α σ)(π) = ϕ α (σ π) = = { }{ } σ π α α α = { } σ α ϕ α (π). α=α α Theorem 5 For each α S, ϕ α is a representative function. α=α α This result is a generalization of Lemma (iii) of [15], here it is proved that subord functions (functions of the form ( u), for, u ords in an alphabet A) are representative functions. Also, it is ell non that representative functions on Q S is an algebra under pointise product (see [5], chapter 1 ). We recover this result in the Theorem 2.

13 850 Yannic Vargas Acnoledgements The author ould lie to than Christophe Reutenauer and Franco Saliola for their advice and support during the preparation of this paper, and also to the to anonymous referees for their helpful comments. References [1] M. Aguiar, F. Sottile. Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. Adv. Math. 191 no. 2, (2005). [2] K. T. Chen, R. H. Fox, R. C. Lyndon. Free differential calculus, IV - The quotient groups of the loer central series, Ann. Math 65, (1978). [3] S. Eilenberg, S. MacLAne. On the groups H(Π, n), Annals of Mathematics, 58, (1953). [4] J.-Y. Enjalbert, H. N. Minh. Combinatorial study of colored Huritz polyzêtas. Discrete Mathematics. 312, (2012). [5] G. P. Hochschild. Basic theory of algebraic groups and Lie algebras. Graduate Texts in Mathematics, 75. Springer-Verlag, Ne Yor-Berlin (1981). [6] M. Hoffman. Quasi-Shuffle products. Journal of Algebraic Combinatorics. 11, (2000). [7] S. A. Joni, G.C. Rota. Coalgebras and bialgebras in combinatorics. Umbral calculus and Hopf algebras (Norman, Ola., 1978), pp. 1-47, Contemp. Math., 6, Amer. Math. Soc., Providence, R.I., (1982) [8] D. Knuth. The Art of Computer Programming, Vol. 1, Fundamental Algorithms, Addison-Wesley, Reading, Massachusetts (1968). [9] M. Lothaire. Combinatorics on ords, Encyclopedia of Mathematics, Vol 17. Addison-Wesley, Reading, MA (1983). [10] C. Malvenuto, C. Reutenauer. Duality beteen quasi-symmetric functions and the Solomon descent algebra. J. Algebra. 177, 3, (1995). [11] K. Neman, D. Radford. The cofree irreducible Hopf algebra on an algebra. Amer. J. Math. 101, no. 5, (1979). [12] P. Ochsenschläger. Binomialoeffizzenten und Shuffle-Zahlen, Technischer Bericht, Fachbereich Informati, T. H. Darmstadt (1981). [13] S. Poirier, C. Reutenauer. Algèbres de Hopf de tableaux. Annales des Sciences Mathématiques du Québec 19, no. 1, (1995). [14] D. E. Radford. A natural ring basis for the shuffle algebra and an application to group schemes. J. Algebra 58, (1979). [15] C. Reutenauer. Free Lie algebras, The Clarendon Press Oxford University Press (1993). [16] Y. Vargas. Hopf algebra of permutation pattern functions (2014).