The # product in combinatorial Hopf algebras

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The # product in combinatorial Hopf algebras

Jean-Christophe Aval, Jean-Christophe Novelli, Jean-Yves Thibon

To cite this version:

Jean-Christophe Aval, Jean-Christophe Novelli, Jean-Yves Thibon. The # product in combinatorial

Hopf algebras. 23rd International Conference on Formal Power Series and Algebraic Combinatorics

(FPSAC 2011), 2011, Reykjavik, Iceland. pp.75-86. �hal-00501646v2�

The # product in combinatorial Hopf algebras

J.-C. Aval

, J.-C. Novelli

and J.-Y. Thibon

1LaBRI, Universit´e de Bordeaux, Talence, France

2Institut Gaspard Monge, Universit´e Paris-Est Marne-la-Vall´ee, Marne-la-Vall´ee, France

Abstract.We show that the#product of binary trees introduced by Aval and Viennot (2008) is in fact defined at the level of the free associative algebra, and can be extended to most of the classical combinatorial Hopf algebras.

R´esum´e.Nous montrons que le produit#introduit par Aval et Viennot (2008) est d´efini au niveau de l’alg`ebre associative libre, et peut ˆetre ´etendu `a la plupart des alg`ebres de Hopf combinatoires classiques.

Keywords:combinatorial Hopf algebras,#product, binary trees, permutations, Young tableaux

1 Introduction

There is a well-known Hopf algebra structure, due to Loday and Ronco [8], on the set of planar binary trees. Using a new description of the product of this algebra, (denoted here byPBT) in terms of Catalan alternative tableaux, Aval and Viennot [1] introduced a new product, denoted by#, which is compatible with the original graduation shifted by 1. Since then, Chapoton [2] has given a functorial interpretation of this operation.

Most classical combinatorial Hopf algebras, includingPBT, admit a realization in terms of special families of noncommutative polynomials. We shall see that on the realization, the#product has a simple interpretation. It can in fact be defined at the level of words over the auxiliary alphabet. Then, it preserves in particular the algebras based on parking functions (PQSym), packed words (WQSym), permuta- tions (FQSym), planar binary trees (PBT), plane trees (the free tridendriform algebraTD), segmented compositions (the free cubical algebraTC), Young tableaux (FSym), and integer compositions (Sym).

All definitions not recalled here can be found,e.g., in [11, 12, 13].

In this extended abstract, most results are presented without proof.

2 A semigroup of paths

LetAbe an alphabet. Words overAcan be regarded as encoding paths in a complete graph with a loop on each vertex, vertices being labelled byA.

Composition of paths, denoted by#, endows the setΣ(A) = A⁺∪ {0}with the structure of a semi- group:

ua#bv=

(uav ifb=a,

0 otherwise. (1)

1365–8050 c2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

For example,baaca#adb=baacadbandab#cd= 0. Thus, the#product mapsAⁿ×A^mtoA^m+n−1. It is graded w.r.t. the path length (i.e., the number of edges in the path).

We have the following obvious compatibilities with the concatenation product:(uv)#w=u·(v#w) and(u#v)·w=u#(vw).

Letd_kbe the linear operator on the free associative algebraKhAi(over some fieldK) defined by dk(w) =

(uav ifw=uaavfor somea, with|u|=k−1,

0 otherwise. (2)

Then, foruof lengthk,u#v=dk(uv).

3 Application to combinatorial Hopf algebras

The notion of acombinatorial Hopf algebrais a heuristic one, referring to rich algebraic structures arising naturally on the linear spans of various families of combinatorial objects. These spaces are generally endowed with several products and coproducts, and are in particular graded connected bialgebras, hence Hopf algebras.

The most prominent combinatorial Hopf algebras can be realized in terms of ordinary noncommutative polynomials over an auxiliary alphabetA. This means that their products, which are described by combi- natorial algorithms, can be interpreted as describing the ordinary product of certain bases of polynomials in an underlying totally ordered alphabetA={a1< a2< . . .}.

We shall see that all these realizations are stable under the#product. In the case ofPBT(planar binary trees), we recover the result of Aval and Viennot [1]. In this case, the#-product has been interpreted by Chapoton [2] in representation theoretical terms.

We shall start with the most natural algebra,FQSym, based on permutations. It contains as subalge- brasPBT(planar binary trees or the Loday-Ronco algebra, the free dendriform algebra on one genera- tor),FSym(free symmetric functions, based on standard Young tableaux), andSym(noncommutative symmetric functions).

It is itself a subalgebra ofWQSym, based on packed words (or set compositions), in which the role ofPBTis played by the free dendriform trialgebra on one generatorTD(based on Schr¨oder trees), the free cubical trialgebraTC(segmented compositions).

Finally, all of these algebras can be embedded inPQSym, based on parking functions.

Note that although all our algebras are actually Hopf algebras, the Hopf structure does not play any role in this paper.

4 Free quasi-symmetric functions: FQSym and its subalgebras

4.1 Free quasi-symmetric functions 4.1.1 The operation d

on FQSym

Recall that the alphabetAistotally ordered. Thus, we can associate to any word overAa permutation σ = std(w), thestandardized wordstd(w)of w, obtained by iteratively scanningwfrom left to right, and labelling1,2, . . .the occurrences of its smallest letter, then numbering the occurrences of the next one, and so on. For example,std(365182122) = 687193245.

For a permutationσ, define

Gσ= X

std(w)=σ

w . (3)

We shall need the following easy property of the standardization:

Lemma 4.1 Let=u1u2· · ·un be a word overA, andσ=σ1σ2. . . σn = std(u). Then, for any factor ofu:std(uiui+1· · ·uj) = std(σiσi+1· · ·σj).

This implies thatFQSymis stable under thed_k: dk(Gσ) =

(Gstd(σ₁···σk−1σ_k+1···σn) ifσk+1=σk+ 1,

0 otherwise. (4)

We shall make use of the dual basis of theGσwhen dealing with subalgebras ofFQSym. In the dual basisFσdefined byFσ :=G_σ⁻¹, the formula is

d_k(F_σ) =

(F_std(σ

1···k[k+1···σn) ifσhas a factork k+1,

0 otherwise, (5)

wherebameans thatais removed.

4.1.2 Algebraic structure

TheG_σspan a subalgebra of the free associative algebra, denoted byFQSym. The product is given by G_αG_β= X

γ=uv,std(u)=α,std(v)=β

G_γ. (6)

The set of permutations occuring in the r.h.s. is called the convolution ofαandβ, and denoted byα∗β.

Hence,

Gσ#Gτ =dk(GσGτ) = X

ν∈σ#τ

Gν, (7)

whereσ#τ={ν| |ν|=k+l−1,std(ν₁. . . ν_k) =σ; std(ν_k. . . ν_k+l−1) =τ}.

Indeed,G_σ#G_τ is the sum of all words of the formw =uxv, withstd(ux) =σandstd(xv) =τ.

For example,G₁₃₂#G₂₃₁=G₁₄₃₅₂+G₁₅₃₄₂+G₂₄₃₅₁+G₂₅₃₄₁.

4.1.3 Multiplicative bases

The multiplicative basesS^σandE^σofFQSymare defined by (see [4]) S^σ=X

τ≤σ

Gτ and E^σ=X

τ≥σ

Gτ, (8)

where≤denotes the left weak order.

Forα∈Sk andβ∈Sl, defineα↑β ∈S_k+l−1as the output of the following algorithm:

• scan the letters ofαfrom left to right and write:αi+β1−1if αi≤αkorαi+ max(β)−1if αi>

α_k,

• scan the letters ofβstarting from the second one and write:β_iif β_i< β₁orβ_i+α_k−1if β_i≥β₁. Similarly, defineα↓βby:

• scan the letters ofαand write:α_iif α_i< α_korα_i+β₁−1if α_i≥α_k,

• scan the letters ofβ starting from the second one and write: βi+αk −1if βi ≤ β1 or βi+ max(α)−1if βi> β1.

For example,3412↑35124 = 78346125and3412↓35124 = 56148237.

Theorem 4.2 The permutations appearing in a#-productGα#Gβis an interval of the left weak order:

Gα#Gβ= X

γ∈[α↓β,α↑β]

Gγ. (9)

Using only either the lower bound or the upper bound, one obtains:

Corollary 4.3 The basesS^σandE^σare multiplicative for the#-product:

S^α#S^β=S^α↑β and E^α#E^β=E^α↓β. (10) For example,S³⁴¹²#S³⁵¹²⁴=S^78346125andE³⁴¹²#E³⁵¹²⁴=E^56148237.

4.1.4 Freeness

The above description of the#product in theSbasis implies the following result:

Theorem 4.4 For the# product,FQSymis free on eitherS^αorGαwhereαruns overnon-secable permutations, that is, permutations of sizen≥2such that any prefixα₁. . . α_kof size2≤k < nis not, up to order, the union of an interval with maximal valueσ_k and another interval either empty or with maximal valuen.

The generating series of the number of non-secable permutations (by shifted degreed⁰(σ) =n−1for σ∈Sn) is Sequence A077607 of [15]

NI(t) := 2t+ 2t²+ 8t³+ 44t⁴+ 296t⁵+ 2312t⁶+. . . (11) or equivalently1/(1−NI(t)) =P

n≥1n!tⁿ⁻¹.

4.2 Young tableaux: FSym

The algebraFSymof free symmetric functions [3] is the subalgebra ofFQSymspanned by the coplactic classes

S_t= X

Q(w)=t

w= X

P(σ)=t

F_σ (12)

where(P, Q)are theP-symbol andQ-symbol defined by the Robinson-Schensted correspondence. This algebra is isomorphic to the algebra of tableaux defined by Poirier and Reutenauer [14]. We shall denote bySTab(n)the standard tableaux of sizen.

Note thatS_{3 4}

1 2

= G₂₄₁₃+G₃₄₁₂, so thatd₁(S_{3 4}

1 2

) = G₃₁₂, which does not belong toFSym.

HenceFSymis not stable under thed_k. However, we have:

Theorem 4.5 FSymis stable under the#-product.

For example,

S₂

1 3

#S₃

2 1

= S₅

4 2 1 3

+S₅

2 4 1 3

(13)

S₃

1 2

#S₃

2 1

= S₅

4 3 1 2

(14)

S₃

2 1

#S₂

1 3

= S₄

3 2 1 5

(15)

S₃

2 1

#S₃

1 2

= S₃

2 5 1 4

+S₅

3 2 1 4

(16)

Note that those products do not have same number of terms, so that there is no natural definition of what would be the#product on the usual (commutative) symmetric functions.

ForTan injective tableau andSa subset of its entries, let us denote byT_|Sthe (sub-)tableau consisting of the restriction ofTto its entries inS. ForT, T⁰two skew tableaux, we denote their plactic equivalence (as for words) byT ≡T⁰, that is we can obtainT⁰ fromT by playingJeu de Taquin. The#product in FSymis given by the following simple combinatorial rule.

Proposition 4.6 LetT1andT2be two standard tableaux of sizeskand`. Then ST₁#ST₂=X

ST (17)

whereTruns over standard tableaux of sizek+`−1such that:T_|{1,...,k}=T1andT|{k,...,k+l−1}≡T2. With this description, it is easy to compute by hand

S₄

1 2 3

#S₃

1 2

=S₆

4 5 1 2 3

+ S₆

4 1 2 3 5

+ S_{4 6}

1 2 3 5

. (18)

4.3 Planar binary trees: PBT 4.3.1 Algebraic structure

Recall that the natural basis ofPBTcan be defined by PT = X

T(σ)=T

Gσ (19)

whereT(σ)is the shape of the decreasing tree ofσ.

Proposition 4.7 The image of a tree byd_kis either0or a single tree:

dk(PT) = (PT⁰,

0, (20)

according to whetherkis the left child ofk+ 1in the unique standard binary search tree of shapeT (equivalently if thek-th vertex in the infix reading ofT has no right child), in which caseT⁰is obtained fromT by contracting this edge, the result being0otherwise.

By the above result, any productPT⁰#PT⁰⁰is inPBT. We just need to select those linear extensions which are not annihilated bydk. Sincedk(Fσ)is nonzero iffσhas (as a word) a factork k+1, the image underdk of the surviving linear extensions are precisely those of the poset obtained by identifying the rightmost node ofT⁰with the leftmost node ofT⁰⁰. Thus,#is indeed the Aval-Viennot product.

4.3.2 Multiplicative bases

The multiplicative basis of initial intervals [5] (corresponding to the projective elements of [2]) is a subset of theSbasis ofFQSym:

HT =S^τ (21)

whereτis the maximal element of the sylvester classT[5]. These maximal elements are the132-avoiding permutations. Hence, they are preserved by the#operation, so that we recover Chapoton’s result: the

#product of two projective elements is a projective element. One can also apply the argument the other way round: since one easily checks that the#product of two permutations avoiding the pattern132also avoids this pattern, it is a simple proof thatPBTis stable under#.

As in the case ofFQSym, the fact that theSbasis is still multiplicative for the#product implies that the product in thePbasis is an interval in the Tamari order.

4.4 Noncommutative symmetric functions: Sym 4.4.1 Algebraic structure

Recall thatSymis freely generated by the noncommutative complete functions Sn(A) = X

i1≤i2≤...≤in

ai1ai2· · ·ain=G12···n (22)

Here, we have obviouslyS_n#S_m=S_n+m−1. This implies, forl(I) =r,I=I⁰i_randJ =j₁J⁰⁰, S^I#S^J =S^{I0·(ir+j1−1)·J00} and similarly R_I#R_J=R_I0·(i_r+j₁−1)·J⁰⁰, (23) whereS^I is the basis of products of noncommutative complete symmetric functions andRI are noncom- mutative ribbon Schur functions. For example,R1512#R43=R15153.

Clearly, as a#-algebra,Sym⁺is the free graded associative algebraKhx, yiover the two generators x=S₂=R₂andy= Λ₂=R₁₁of degree1, the neutral element beingS₁=R₁= Λ₁.

Now, define for any compositionI= (i0, . . . , ir), the binary word:b(I) = 0ⁱ⁰⁻¹1(0ⁱ¹⁻¹)1. . .(0^ir−1).

On the binary coding of a composition I, one can read an expression of R_I, S^I, and Λ^I in terms of

#-products of the generators x, y: replace the concatenation product by the #-product, replace0 by respectivelyx,x, ory, and1by respectivelyy,x+y, orx+y, so that

R_I := (xⁱ⁰⁻¹)^##y#(xⁱ¹⁻¹)^##y#. . .#(x^ir−1)^#, (24) S^I := (xⁱ⁰⁻¹)^##(x+y)#(xⁱ¹⁻¹)^##(x+y)#. . .#(x^ir−1)^#, (25) and

Λ^I := (yⁱ⁰⁻¹)^##(x+y)#(yⁱ¹⁻¹)^##(x+y)#. . .#(y^ir−1)^#. (26) Note that in particular, the maps sending S^I either to RI or Λ^I are algebra automorphisms. This property will extend to a Hopf algebra automorphism with the natural coproduct.

4.4.2 Coproduct

In this case, we have a natural coproduct: the one for whichxandyare primitive:

∇S₂=S₂⊗S₁+S₁⊗S₂, and ∇Λ₂= Λ₂⊗Λ₁+ Λ₁⊗Λ₂, (27) and, the neutral elementS₁is grouplike:∇S₁=S₁⊗S₁. Then,

∇Sn=

n

X

i=1

n−1 i−1

Si⊗Sn+1−i. (28)

The coproduct of genericS^I,Λ^I, andRI all are the same: sincexandy are primitive,x+yis also primitive, so that,e.g.,

∇RI = X

w,w⁰|w w⁰=b(I)

C_w,w^b(I)0RJ⊗RK, (29)

whereJ(resp.K) are the compositions whose binary words arew(resp.w⁰), andC_w,w^b(I)0is the coefficient ofb(I)inw w⁰. Another way of presenting this coproduct is as follows: givenb(I), choose for each element if it appears on the left or on the right of the coproduct (hence giving2^|I|−1terms) and compute the corresponding products ofxandy.

Hence, the maps sendingS^I either toR_I orΛ^I are Hopf algebra automorphisms.

4.4.3 Duality: quasi-symmetric functions under #

SinceSymis isomorphic to the Hopf algebraKhx, yion two primitive generatorsxandy, its dual is the shuffle algebra on two generators whose coproduct is given by deconcatenation.

Since all three basesS,R, andΛbehave in the same way for the Hopf structure, the same holds for their dual bases, so that the basesMI,FI, and the forgotten basis ofQSym have the same product and coproduct formulas. In the basisFI, this is

F_I#F_J = X

w∈b(I) b(J)

F_K, (30)

whereKis the composition such thatb(K) =w.

For example,F3#F12= 3F14+ 2F23+F32, sincexx yx= 3yxxx+ 2xyxx+xxyx.

Note that since the product is a shuffle on words inxandy, all elements in a productFIFJhave same length, which isl(I) +l(J)−1.

5 Word quasi-symmetric functions: WQSym and its subalgebras

5.1 Word quasi-symmetric functions 5.1.1 Algebraic structure

Word quasi-symmetric functions are the invariants of the quasi-symmetrizing action of the symmetric group (in the limit of an infinite alphabet), see,e.g., [12].

Thepacked wordu = pack (w)associated with a wordwin the free monoidA^∗ is obtained by the following process. Ifb1 < b2 < . . . < br are the letters occuring in w,uis the image of wby the homomorphismbi 7→ai. For example,pack (469818941) = 235414521.A worduis said to bepacked ifpack (u) =u. Such words can be interpreted as set compositions, or as faces of the permutohedreon, and are sometimes called pseudo-permutations [6].

As in the case of permutations, we have:

Lemma 5.1 Letu = u1u2· · ·un be a word overA, andv = v1v2. . . vn = pack (u). Then, for any factor ofu,pack (uiui+1· · ·uj) = pack (vivi+1· · ·vj).

The natural basis ofWQSym, which lifts the quasi monomial basis ofQSym, is labelled by packed words. It is defined by

Mu= X

pack (w)=u

w . (31)

Note thatWQSymis stable under the operatorsdk. We have dk(Mw) =

(Mw₁···wk−1w_k+1···w_n ifwk=wk+1,

0 otherwise, (32)

so thatWQSymis also stable under#. Since in this basis the ordinary product is given by M_uM_v= X

w=u⁰v⁰; pack (u⁰)=u,pack (v⁰)=v

M_w, (33)

we have

M_u#M_v=d_k(M_uM_v) = X

w∈u#v

M_w, (34)

whereu#v={w| |w|=k+l−1,pack (w₁. . . w_k) =u; pack (w_k. . . w_k+l−1) =v}.

For example,M₁₂₁#M₁₂=M₁₂₁₂+M₁₂₁₃+M₁₃₁₂.

5.1.2 Multiplicative bases

Recall that there exists an order on packed words generalizing the left weak order : it is the pseudo- permutohedron order. This order has a definition in terms of inversions (see [6]) similar to the definition of the left weak order. Thegeneralized inversion setof a given packed wordwis the union of the set of pairs(i, j)such thati < j andwi > wj with coefficient one (full inversions), and the set of pairs(i, j) such thati < jandwi=wjwith coefficient one half (half-inversions).

One then says that two wordsuandvsatisfyu < vfor the pseudo-permutohedron order iff the coeffi- cient of any pair(i, j)inuis smaller than or equal to the same coefficient inv.

Note that the definition ofu↑vandu↓v(see the section aboutFQSym) does not requireuandvto be permutations. One then has

Theorem 5.2 The words appearing in the productM_u#M_vis an interval of the pseudo-permutohedron order:

M_u#M_v= X

w∈[u↓v,u↑v]

M_w. (35)

The multiplicative basesS^uandE^uofWQSymare defined in [12] by S^u=X

v≤u

M_v and E^u=X

v≥u

M_v, (36)

where≤is the pseudo-permutohedron order.

Proposition 5.3 TheSandE-bases are multiplicative for the#-product:

S^u#S^v=S^u↑v and E^u#E^v=E^u↓v. (37)

5.1.3 Freeness

As in the case ofFQSym, we can describe a set of free generators in theSbasis forWQSym.

We shall say that a packed worduof sizenissecableif there exists a prefixu₁. . . u_kof size2≤k < n such that:{u1, . . . , uk} ∩ {uk, . . . , un}={uk}and the set{u1, . . . , uk}is, up to order the union of an interval with maximal valueuk and another interval either empty or with maximal value the maximal entry of the whole wordu.

Conversely, a packed word of size at least2which is not secable will be callednon-secable.

Theorem 5.4 For the# product,WQSym is free on theS^u orMu whereuruns over non-secable packed words.

If a packed worduis weighted byt^|u|−1, the generating seriesPW(t)of (unrestricted) packed words corresponds to Sequence A000670 of [15]:

PW(t) = 1 + 3t+ 13t²+ 75t³+ 541t⁴+ 4683t⁵+ 47293t⁶+. . . (38) The generating seriesN SP W of non-secable packed words is related toP W by the relationPW(t) = 1/(1−N SP W(t))which enables us to computeN SP W:

NSPW(t) = 3t+ 4t²+ 24t³+ 192t+ 1872t⁵+ 21168t⁶+. . . (39)

5.2 The free tridendriform algebra TD

The realization of the free dendriform trialgebra given in [11] involves the following construction. With any wordwof lengthn, associate a plane treeT(w)withn+ 1leaves, as follows: ifm= max(w)and ifwhas exactlyk−1occurences ofm, write

w=v1m v2· · ·v_k−1m vk, (40) where thevimay be empty. Then,T(w)is the tree obtained by grafting the subtreesT(v1), , . . . ,T(vk) (in this order) on a common root, with the initial conditionT() =∅for the empty word. For example, the tree associated with243411is

xxxxx KKKKKK ,,, 4

,,, 4 8888

2 3 1 1

(41)

We shall callsectorsthe zones containing numbers and say that a sector is to the left of another sector if its number is to the left of the other one, so that the reading of all sectors from left to right of anyT(w) gives backw.

Now define a polynomial by

MT := X

T(w)=T

Mw. (42)

Then, exactly as in the case ofPBT, we have Theorem 5.5

dk(MT) =

MT⁰,

0, (43)

depending on whether thek-th andk+ 1-th sectors are grafted on the same vertex or not. In the nonzero case,T⁰is obtained fromTby gluing thek-th andk+1-th sectors.

Corollary 5.6 The operation#is internal onTD.

5.3 The free cubical algebra TC

Define asegmented compositionas a finite sequence of integers, separated by vertical bars or commas,e.g., (2,1|2|1,2). We shall associate an ordinary composition with a segmented composition by replacing the vertical bars by commas. Recall that the descents of an ordinary composition are the positions of the ones in the associated binary word. In the same way, there is a natural bijection between segmented compositions of sumnand sequences of lengthn−1over three symbols<,=, >: start with a segmented compositionI. Ifiis not a descent of the underlying composition ofI, write<; otherwise, ificorresponds to a comma, write=; ificorresponds to a bar, write>.

Now, with each wordwof lengthn, associate a segmented compositionS(w), defined as the sequence s1, . . . , s_n−1wheresiis the comparison sign betweenwiandwi+1.

For example, givenw = 1615116244543, one gets the sequence<><>=<><=<>>and the seg- mented composition(2|2|1,2|2,2|1|1).

Given a segmented compositionI, define:MI=P

S(T)=IMT.

It has been shown in [12] that theMIgenerate a Hopf subalgebra ofTDand that their product is given by

MI⁰MI⁰⁰ =MI⁰.I⁰⁰+MI⁰,I⁰⁰+MI⁰|I⁰⁰. (44) whereI⁰.I⁰⁰is obtained by gluing the last part ofI⁰with the first part ofI⁰⁰.

As before, it is easy to see that Theorem 5.7

dk(MI) =

MI⁰,

0, (45)

depending on whetherkis not or is a descent of the underlying composition ofI. In the nonzero case, I⁰ is obtained fromIby decreasing the entry that corresponds to the entry containing thek-th cell in the corresponding composition, that is, ifI= (i₁, . . . , i_`)where theiare separated by commas or vertical bars, decreasingi_nwherenis the smallest integer such thati₁+· · ·+i_n> k.

For the same reason, the following result is also true:MI⁰#MI⁰⁰ =MI⁰.⁰I⁰⁰, whereI⁰.⁰I⁰⁰amounts to glue together the last part ofI⁰with the first part ofI⁰⁰minus one, leaving the other parts unchanged.

6 Parking quasi-symmetric functions: PQSym

Aparking functionon[n] = {1,2, . . . , n}is a worda = a₁a₂· · ·a_n of length non[n] whose non- decreasing rearrangementa^↑ = a⁰₁a⁰₂· · ·a⁰_n satisfiesa⁰_i ≤ifor alli. We shall denote byPFthe set of parking functions.

For a wordwover a totally ordered alphabet in which each element has a successor, one can define [13] a notion ofparkized wordpark(w), a parking function which reduces tostd(w)whenwis a word without repeated letters.

Forw=w1w2· · ·wnon{1,2, . . .}, we set

d(w) := min{i|#{wj ≤i}< i}. (46)

Ifd(w) = n+ 1, thenwis a parking function and the algorithm terminates, returningw. Otherwise, letw⁰be the word obtained by decrementing all the elements ofwgreater thand(w). Thenpark(w) :=

park(w⁰). Sincew⁰ is smaller thanwin the lexicographic order, the algorithm terminates and always returns a parking function.

For example, letw= (3,5,1,1,11,8,8,2). Thend(w) = 6and the wordw⁰= (3,5,1,1,10,7,7,2).

Thend(w⁰) = 6andw⁰⁰= (3,5,1,1,9,6,6,2). Finally,d(w⁰⁰) = 8andw⁰⁰⁰ = (3,5,1,1,8,6,6,2), that is a parking function. Thus,park(w) = (3,5,1,1,8,6,6,2).

Lemma 6.1 Letu=u1u2· · ·unbe a word overA, andc=c1c2. . . cn= park(u). Then, for any factor ofu:park(uiui+1· · ·uj) = park(cici+1· · ·cj).

Recall from [13], that with a parking functiona, one associates the polynomial Ga= X

park(w)=a

w . (47)

These polynomials form a basis of a subalgebra⁽ⁱ⁾PQSymof the free associative algebra overA. In this basis, the product is given by

GaGb= X

c=uv; park(u)=a,park(v)=b

Gc. (48)

Thus,

Ga#Gb= X

c∈a#b

Gc, (49)

wherea#b={c| |c|=k+l−1,park(c1. . .ck) =a,park(ck. . .c_k+l−1) =b}. Indeed,Ga#Gbis the sum of all words of the formw=uxv, withpark(ux) =aandpark(xv) =b.

Note thatPQSymis not stable under the operatorsdk. For example,d1(G112)is not inPQSym.

However, letd⁰_kbe the linear operator defined by d⁰_k(G_c) =

(Gc₁...c_k−1c_k+1...c_n ifck =ck+1andc1. . .c_k−1ck+1. . .cn∈PF,

0 otherwise. (50)

(i)Strictly speaking, this subalgebra is ratherPQSym^∗, the graded dual of the Hopf algebraPQSym, but both are actually isomorphic.

Proposition 6.2 Then, ifais of lengthk, one has:G_a#G_b=d⁰_k(G_aG_b). For example,G₁₂₁#G₁₁₄₁=G₁₂₁₁₆₁+G₁₂₁₁₅₁+G₁₂₁₁₄₁.

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