Bell Polynomials in Combinatorial Hopf Algebras Ammar Aboud, Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, Olivier Mallet

Bell polynomials in combinatorial Hopf algebras Ammar Aboud, Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, Olivier Mallet

To cite this version:

Ammar Aboud, Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, Olivier Mallet. Bell polynomials in combinatorial Hopf algebras. 2014. hal-00945543v1

HAL Id: hal-00945543 https://hal.archives-ouvertes.fr/hal-00945543v1 Preprint submitted on 12 Feb 2014 (v1), last revised 21 Jan 2016 (v3)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Bell polynomials in combinatorial Hopf algebras

Ammar Aboud,∗ Jean-Paul Bultel,† Ali Chouria‡ Jean-Gabriel Luque§and Olivier Mallet¶

February 12, 2014

Keywords: Bell polynomials, Symmetric functions, Hopf algebras, Faà di Bruno algebra, Lagrange inversion, Word symmetric functions. Abstract Partial multivariate Bell polynomials have been deﬁned by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities etc. Many of the formulæ on Bell polynomials involve combinatorial objects (set partitions, set partitions in lists, permutations etc). So it seems natural to investigate analogous formulæ in some combinatorial Hopf algebras with bases indexed by these objects. The algebra of symmetric functions is the most famous example of a combinatorial Hopf algebra. In a ﬁrst time, we show that most of the results on Bell polynomials can be written in terms of symmetric functions and transformations of alphabets. Then, we show that these results are clearer when stated in other Hopf algebras (this means that the combinatorial objects appear explicitly in the formulæ). We investigate also the connexion with the Faà di Bruno Hopf algebra and the Lagrange-Bürman formula.

1 Introduction

Partial multivariate Bell polynomials (Bell polynomials for short) have been deﬁned by E.T. Bell in [3] in 1934. But their name is due to Riordan [28] which studied the Faà di Bruno formula [14, 15] allowing one to write the nth derivative of a composition f ◦ g in terms of the derivatives of f and g [27]. The applications of Bell polynomials in Combinatorics, Analysis, Algebra, Probabilities etc. are so numerous that it should be very long to detail them in the paper. Let us give very few seminal examples. • The main applications to Probabilities follow from the fact that the nth moment of a probability distribution is a complete Bell polynomial of the cumulants.

∗[email protected]. Ecole Nationale Supérieure d’Informatique BP 68M OUED SMAR, 16270, EL HARRACH ALGER ALGERIE †[email protected]. Laboratoire LITIS - EA 4108, Université de Rouen, Avenue de l’Université - BP 8 76801 Saint-Étienne-du-Rouvray Cedex ‡[email protected]. Laboratoire LITIS - EA 4108, Université de Rouen, Avenue de l’Université - BP 8 76801 Saint-Étienne-du-Rouvray Cedex §[email protected]. Laboratoire LITIS - EA 4108, Université de Rouen, Avenue de l’Université - BP 8 76801 Saint-Étienne-du-Rouvray Cedex ¶[email protected]. Laboratoire LITIS - EA 4108, Université de Rouen, Avenue de l’Université - BP 8 76801 Saint-Étienne-du-Rouvray Cedex

1 • Partial Bell polynomials are linked to the Lagrange inversion. This follows from the Faà di Bruno formula. • Many combinatorial formulæ on Bell polynomials involve classical combinatorial numbers like Stirling numbers, Lah numbers etc. The Faà di Bruno formula and many combinatorial identities can be found in [10]. The PhD thesis of M. Mihoubi [26] contains a rather complete survey of the applications of these polynomials together with numerous formulæ. Some of the simplest formulæ are related to the enumeration of combinatorial objects (set partitions, set partitions into lists, permutations etc.). So it seems natural to investigate analogous formulæ in some combinatorial Hopf algebras with bases indexed by these objects. Combinatorial Hopf algebras are graded bialgebras with bases indexed by combinatorial objects such that the product and the coproduct have some compatibilities. The graduation implies that the Hopf structure is equivalent to the fact that the coproduct is a morphism for the product. In Section 2, we recall some facts about two important examples of Hopf algebras: the algebra of symmetric functions and the algebra of word symmetric functions. The most studied example is the Hopf algebra of symmetric functions Sym. The importance of this algebra is due to its applications in the representation theory of the symmetric group (see e.g. [23]). The algebra Sym can also be used to encode equalities on generating functions via the notion of specialization of alphabet (see e.g. [22]). By interpreting Bell polynomials in the context of symmetric functions, some identities involving these polynomials are recovered from well known relations involving different bases of symmetric functions. Indeed, by identifying the exponential generating function of the entries with a Cauchy function, we give an expression of the Bell polynomials in terms of complete symmetric functions. So, in Section 3, we use properties of symmetric functions to prove already known identities about Bell polynomials as well as some new ones. There are many ways to construct a Hopf algebra from an algebra since several coproducts can be defined to have the desired property. For instance, the Faà di Bruno algebra is another Hopf algebra based on the algebra Sym. This algebra has strong links with the Faà di Bruno formula and the Lagrange inversion. The Bell polynomials appear naturally in this context. The aim of Section 4 is to investigate properties of Bell polynomials with respect to the composition of functions. Word symmetric polynomials form a Hopf algebra with bases indexed by set partitions. This algebra is an ideal candidate to define an analogue of Bell polynomials. In Section 5, we define word Bell polynomials and investigate many of their properties. Finally in Section 6, we investigate analogues of Bell polynomials in other combinatorial Hopf algebras.

2 Symmetric and word symmetric functions

In this section we describe two combinatorial Hopf algebras that we will link to Bell polynomials in the next sections: Sym and WSym. Both of them are deﬁned as the set of polynomials invariant under permutation of the variables.

2.1 The algebra of symmetric functions The algebra of symmetric functions is formally deﬁned as the free algebra over an inﬁnite number of symbols pi, i ∈ N \{0} (we usually set p0 = 1). The bases of Sym are indexed by the partitions

2 λ ⊢ n of all the integers n. A partition λ of n is a finite noncreasing sequence of positive integers (λ1 ≥ λ2 ≥ ...) such that i λi = n. The multiplicity mj(λ) of j in λ ⊢ n is the number of parts of λ equal to j, the length ℓ(λ) of λ is its number of parts, and n is the weight |λ| of λ. We set mi(λ) P also zλ = i mi(λ)!i . Let pλ := pλ1 ··· pλk for any partition λ = [λ1, . . . , λk]. The scalar product defined by hpλ, pµi = zλδλ,µ, where δλ,µ = 1 if λ = µ and 0 otherwise, comes from the representation Q theory of the symmetric group and allows us to define an autodual Hopf algebra structure. We will use the notations of [23]. Alternatively, the algebra Sym is known to be isomorphic to its polynomial realization Sym(X) on an infinite set X = {x1, x2,...} of commuting variables. The algebra Sym(X) is the space of the polynomials that are invariant under permutation of the variables. We identify any symmetric function f with its standard realisation f(X) when convenient. The nth power sum X X n symmetric function pn( ) is defined by pn( ) = i xi , and the nth complete symmetric function hn(X) is the sum of all the monomials of degree n on the elements of X. These two free families of P generators of Sym are linked by the Newton formula tn σ (X) = exp{ p (X) }, (1) t n n n>1 X X X X X n where σt( ) is the generating function of the hn( ), called Cauchy function: σt( ) = n≥0 hn( )t . From the pn and the hn, one defines the multiplicative bases (pλ)λ and (hλ)λ of Sym(X), setting X X X X X P pλ( ) = i pλi ( ) and hλ( ) = i hλi ( ). For ℓ(λ) ≤ p and = (x1, . . . , xp), we set

Q Q λ1+p−1 λ1+p−2 λp x1 x1 . . . x1 λ1+p−1 λ1+p−2 λp x2 x2 . . . x2 A (X) = . (2) λ ...... λ1+p−1 λ1+p−2 λp xp xp . . . xp

Then, the Schur function sλ deﬁned by

Aλ(X) sλ(X) := (3) 1≤i

∆(f) = f(X + Y), (6)

3 which endows Sym with its classical Hopf structure. Also if α ∈ C the alphabet αX and XY are deﬁned by

pn(αX) = αpn(X) and pn(XY) = pn(X)pn(Y), (7)

Finally, we give the Cauchy formula, which relates the product of two alphabets with the generating kernel of the scalar product on symmetric functions:

1 |λ| σt(XY) = pλ(X)pλ(Y)t . (8) zλ Xλ The reader can refer to [22] for more details about symmetric functions and alphabets.

2.2 The algebra of word symmetric functions The algebra of word symmetric functions is a way to construct a noncommutative analogue of the algebra Sym. Its bases are indexed by set partitions. This algebra appeared ﬁrst in [29] and its name comes from its realization as a subalgebra of ChAi, where A = {a1, . . . , an,... } is an inﬁnite alphabet. π Consider the family Φ := {Φ }π whose elements are indexed by set partitions of {1, . . . , n}. The algebra ′ ′ WSym [29] is generated formally by Φ using the shifted concatenation product: ΦπΦπ = Φππ [n] where π and π′ are set partitions of {1, . . . , n} and {1, . . . , m}, respectively, and π′[n] means that we add n to each integer occurring in π′.

Example 2.1 Φ{1,3}{2}Φ{1,4}{2,5,6}{3,7}{8} = Φ{1,3}{2}{4,7}{5,8,9}{6,10}{11} . π A The polynomial realization is defined by Φ ( ) = w w where the sum is over the words w = a1 ··· an where i, j ∈ πℓ implies ai = aj, if π = {π , . . . , πk} is a set partition of {1, . . . , n}. 1 P {1,4}{2,5,6}{3,7} Example 2.2 Φ (A) = a,b,c∈A abcabbc. π Other bases are known, for example,P the word monomial functions defined by Φ = π≤π′ Mπ′ (where π ≤ π′ indicates that π is finer than π′, i.e., that each block of π′ is a union of blocks of π). P Example 2.3

{1,4}{2,5,6}{3,7} Φ = M{1,4}{2,5,6}{3,7} + M{1,2,4,5,6}{3,7} + M{1,3,4,7}{2,5,6}

+ M{1,4}{2,3,5,6,7} + M{1,2,3,4,5,6,7}.

From the definition of the Mπ, we deduce that the polynomial representation of the word monomial A functions is given by Mπ( ) = w w where the sum is over the words w = a1 ··· an where i, j ∈ πℓ if and only if ai = aj, where π = {π , . . . , πk} is a set partition of {1, . . . , n}. P1 A Example 2.4 M{1,4}{2,5,6}{3,7}( ) = a,b,c∈A abcabbc. a6=b,a6=c,b6=c P Although the construction of WSym(A) seems to be close to Sym(X), their algebraic properties are quite different since the algebra WSym(A) is not autodual. Surprisingly, the graded dual WSym⋆ of WSym admits a realization in the space WSym(A) involving the shuffle product and some bases

4 usually deﬁned in WSym⋆ satisfy combinatorial identities involving bases in WSym. For instance, one can deﬁne a word analogue of complete symmetric functions by

A ′ π′ A Sπ( ) = card(πi) Φ ( ), (9) ′ i ! πX≤π Y ′ ′ ′ where π = {π1, π2,...}. ⋆ Note that the complete basis (Sπ)π of WSym is dual to the monomial basis (Mπ). ∗ For any basis (Bπ) of WSym, we will set Bn := B{{1,...,n}}. The Hopf structure of WSym and WSym has been studied by Bergeron et al. [4]. In the sequel, when there is no ambiguity, we will identify the algebras WSym and WSym(A). Since the subalgebra of WSym generated by the complete functions Sn(A) is isomorphic to Sym, we define W A W A σt ( ) and φt ( ) by W A A n σt ( ) = Sn( )t (10) n X≥0 and W A A n−1 φt ( ) = n!Φn( )t . (11) n X≥1 These series are linked by the equality d φW (A) = log (σW (A)), (12) t dt t where is the shuffle product of the words, and log is the logarithm in (WSym(A), ), that is the algebraic structure on the vector space WSym(A) obtained by replacing the concatenation product of words by the shuffle product. Note that as an associative algebra, (WSym(A), ) is isomorphic to WSym⋆. Indeed, the coproduct of WSym consists of identifying the algebra WSym ⊗ WSym with WSym(A + B), where A and B are two commuting alphabets. Hence, one has W A B W A W B σt ( + ) = σt ( ) σt ( ). (13) The notion of specialization is more subtle to define than in Sym. Indeed, the knowledge of the complete function Sn(A) does not allow us to recover all the polynomials using uniquely the algebraic operations. In [6], we made an attempt to define virtual alphabets by reconstituting the whole algebra using the action of an operad (C-modules). Although the general mechanism remains to be defined, the case where each complete function Sn(A) is specialized to a sum of words of length n can be understood via this construction. More precisely, we consider the family of multilinear k-ary operators 1 1 k k p Π (Π is a set composition) on words by (a ··· a , . . . , a ··· a ) = b1 ··· bn with b p = a [π1,...,πk] 1 n1 nk iℓ ℓ p p B B if πp = {i1 ≤ · · · ≤ ink }. If we consider homogeneous polynomials Sn[ ], we define S{π1,...,πk}[ ] = B B B [π1,...,πk](S#π1 [ ],...,S#πk [ ]). The space generated by the polynomials S{π1,...,πk}[ ] endowed with the product · and is homomorphic to the bi-algebra (WSym(A), ·, ). See [6] and Section 5 for more details.

3 Bell polynomials and symmetric functions

Most of the results contained in the section are already known, but we show that they can be fully understood in terms of symmetric functions and virtual alphabets. We give also a few new results that are diﬃcult to prove without the help of symmetric functions.

5 3.1 Deﬁnitions and basic properties

The (complete) Bell polynomials are usually defined on an infinite set of commuting variables {a1, a2,...} by the following generating function: tn tm A (a , . . . , a ,... ) = exp( a ), (14) n 1 p n! m m! n> m> X0 X1 and the partial Bell polynomials are defined by tn 1 tm B (a , . . . , a ,... ) = ( a )k. (15) n,k 1 p n! k! m m! n>0 m>1 X X

So we have An = k Bn,k. Without loss of generality, we will suppose a1 = 1 in the sequel. Indeed, k a2 ap if a1 6= 0, then the generating function gives Bn,k(a1, . . . , ap,... ) = a Bn,k 1, , ··· , and when P 1 a1 a1 a1 = 0, 0 if n < k B (0, a , . . . , a ,... ) = n,k 2 p n! B (a , . . . , a ,... ) if n ≥ k. ( (n−k)! n,k 2 p These polynomials are related to several combinatorial sequences which involve set partitions. For instance, Bn,k(1, 1,... ) = Sn,k is the Stirling number of the second kind Sn,k (Sloane’s sequence A106800 [30]), which counts the ways to partition a set of n elements into k non empty subsets. n k Note also that An(x, x, . . . ) = k=0 Sn,kx is the classical univariate Bell polynomial denoted by φn(x) in [3]. P

3.2 Bell polynomials as symmetric functions Many equalities on Bell polynomials can be proved by manipulation of the generating functions. In fact, all these computations are more easily understandable when encoded in terms of symmetric functions and specializations. Recall that for our purpose, and without loss of generality, we consider a1 = 1. Consider X a virtual alphabet satisfying ai = i!hi−1(X) for any i ≥ 1 and for simplicity, let hn(X) := n!hn(X). One obtains: e

k k tn 1 a 1 B (a , a ,... ) = i ti = h (X)ti+1 n,k 1 2 n! k! i! k! i n>0 i> ! i> ! X X1 X0 k tk tk tk = h (X)ti = σ (X)k = σ (kX). k! i k! t k! t i> ! X0 Hence, n! n B (1, 2!h ,..., (m + 1)!h (X),... ) = h (kX) = h (kX). (16) n,k 1 m k! n−k k n−k e X n X In the sequel, we will denote by Bn,k the symmetric function deﬁned by Bn,k( ) := k hn−k(k ) for any alphabet X. e 6 Example 3.1 Let us give a few classical examples.

1. The simplest one is given by a1 = 1 and ai = 0 for each i > 1. The specialization satisfying ai = i!hi−1(X) is described by the Cauchy function σt(X) = 1. Hence, for any integer k > 0, one has hn(kX) = 0 if n > 0 and h0(kX) = 1. So

1 if n = k B (X) = n,k 0 otherwise. X 1 X exp(t)−1 2. Set ai = 1 for each i. Hence, we have hi( ) = (i+1)! and σt( ) = t , and we recognize easily Bn,k(X) = Sn,k from the exponential generating function of the original Bell polynomials: 1 n t n n! φn(x)t = exp{x(e − 1)}. Indeed, P 1 1 X k n+k t n+k 1 t k (n+k)! Sn+k,k = (n+k)! Bn+k,k( ) = [x ][t ] exp{x(e − 1)} = [t ] k! (e − 1) k n 1 et−1 1 X = [t ] k! k = k! hn(k ). 3. When ai = i! we need to consider the specialization hi(X) = 1. Hence, the generating function k X X 1 X of hn(k ) is σt(k ) = 1−x . In other words, we have Bn,k(1!, 2!, . . . , m!,... ) = Bn,k( ) = n−1 n! k−1 k! , the Lah number Ln,kwhich counts the number of partitions of the set {1, . . . , n} into k lists. X 1 X 4. If ai = i then hi( ) = i! and σt(k ) = exp(kt), then a straightforward computation gives n n−k Bn,k(1, 2, . . . , m, . . . ) = k k . X X n X tn Comparing (14) and σt( ) = n hn( )t = exp{ n≥1 pn( ) n }, we can consider the complete Bell polynomials An as the complete functions hn(X). p P e P So we will deﬁne An(X) := hn(X) = An(0!p1(X), 1!p2(X),..., (n−1)!pn(X),... ) for any virtual alphabet X. With such a specialization, the partial Belle polynomials read e 1 Bp = B (0!p (X), 1!p (X),..., (n − 1)!p (X),... ) = n! pλ(X). (17) n,k n,k 1 2 n z λ⊢n λ ℓ(Xλ)=k In other words, p X k n X Bn,k( ) = n![α ][t ]σt(α ). (18)

p p Example 3.2 In certain cases, it is easier to consider the polynomials An and Bn,k. For instance, let p X be the virtual alphabet deﬁned by pn(X) = 1 for each n ∈ N. Then, An(X) = An(0!, 1!, 2!,..., (m − α X tn 1 p X k n 1 1)!,... ) = n! since σt( ) = exp{ n! } = 1−t . In the same way, Bn,k( ) = n![α ][t ] 1−t = sn,k, a Stirling number of the ﬁrst kind. P i−1 Example 3.3 A more complicated example is treated in [5, 21] where ai = i . In this case, the X ∞ n−1 tn specialization gives σt(α ) = exp{−αW (−t)} where W (t) = n=1(−n) n! is the Lambert W P

7 α X W (−t) function satisfying W (t) exp{W (t)} = t (see e.g. [11]). Hence, σt(α ) = −t . But the expansion α W (t) of the series t is known to be:

∞ W (t) α 1 = 1 + α(α + n)n−1(−t)n. (19) t n! n=1 X p X n−1 n−k Hence, we obtain Bn,k( ) = k−1 n . Note that the expansion of W (t) and (19) are usually obtained by the use of the Lagrange inversion. We will see in the sequel how this tool is related to our purpose. 3.3 Bell polynomials and binomial functions The partial binomial polynomials are known to be involved in interesting identities on binomial functions. In this section, we explain why the interpretation in terms of symmetric functions allows us to understand soundly these identities. In particular, we are interested by some equalities of [25, 5]; note also that several results on Bell polynomials are collected in [26]. Let us first recall that a binomial sequence is a family of functions (fn)n∈N satisfying f0(x) = 1 and n n f (a + b) = f (a)f (b) (20) n k k n−k Xk=0 for all a, b ∈ C and n ∈ N. This last identity is nothing but the sum of two alphabets stated in terms of modified complete functions hn. Indeed, considering the two specializations given by hn(A) := fn(a) B A B n n A B and hn( ) := fn(a), (20) is equivalent to the classical hn( + ) = k=0 k hk( )hn−k( ) which is easily obtained from σ (A + Be) = σ (A)σ (B) and so f (ka) = h (kA). Hence, as a directe consequence t t t n n P of (16),e we obtain e e e e n n B (1, . . . , if (a),... ) = B (A) = h (kA) = f (ka). (21) n,k i−1 n,k k n−k k n−k X n n X X e Similarly, hn((k1 + k2) ) = i=0 i hi(k1 )hn−i(k2 ) gives Pn n e B = h˜ e ((ek + k )A) = h˜ (k A)h˜ (k A). n,k1+k2 k + k n−k1−k2 1 2 i−k1 1 n−k2−i 2 1 2 i X=0 Hence n k + k n 1 2 B = B B . (22) k n,k1+k2 i i,k1 n−i,k2 1 i X=0 n −1 From (22), we remark that the family of functions (fn)n∈N, defined by fn(k) = k Bn,k if n > 0 and f = 1, is binomial and we obtain 0 n −1 i − 1 −1 n − k −1 B (1, . . . , i B (X),... ) = 1 B . (23) k k n,k1 k i−1,k2 k k n−k1,k1k2 1 2 1 2 Note that (22) is easily generalized as k + ··· + k n 1 p B = B ··· B (24) k , . . . , k n,k1+···+kp i , . . . , i i1,k1 ip,kp 1 p i i n 1 p 1+···X+ p= 8 which is obtained from σt((k1 + ··· kp)X) = σt(k1X) ··· σt(kpX). The extremal case of (24) is

1 n−1 i1−1 ik−1−1 n Bn,k+1 = ··· k+1 i1=k i2=k−1 ik=1 n−i1,i1−i2,...,ik−1−ik (25) (n − i )(i − i ) ··· (i − i )h h ··· h , 1 1 P2 Pk−1 k n−Pi1−1 i1−i2−1 ik−1−ik−1 X which can be obtained directly by considering thee expansione of hn−k(ek ) in the monomial basis. All the identities of this section are already known, for instance (22) and (25) have been proved in [12] and (23) is a special case of Theorem 2.18 in [26]. We have seen that many identities on partial Bell polynomials are derived from the Cauchy series of a sum of alphabets σt(X + Y) = σt(X)σt(Y). Let us conclude this section by giving a last example of such a calculation. Consider the identity (see e.g. [12]):

n−k 1 n n + 1 B = (k + 1) − (i + 1)h B . (26) n,k n − k i i + 1 i n−i,k i X=1 e If we take the coeﬃcient of tn−k−1 in the left hand side and the right hand side of the equality d X d X X dt σt((k + 1) ) = (k + 1) dt σt( ) σt(k ), we obtain

n−k (n − k)hn−k((k + 1)X) = (k + 1) ihi(X)hn−i−k. (27) i X=1 X n−k X X k! Since hn−k((k + 1) ) = i=0 hi( )hn−i−k(k ) and hn−k = n! Bn,k, we obtain after solving (27) as B B 1 n−k n! i (k+1) k! n k 1 k! i h B . an equation in n,k: n,kP = n−k i=1 k! (i+1)! (n−i)! − ( − ) (i+1)! (n−i)! ( + 1)! i n−i,k After simplifying the expression, we recoverP (26) ash expected. i

3.4 Other classical transformations of alphabets 3.4.1 Sums of alphabets, again The results involving binomial functions can be seen as a generalization of the (so-called) convolution formula for Bell polynomials (see e.g. [26]):

n−k m n n 1 m + 1 B (a , a ,... )B (b , b ,... ) = B a b ,..., a b ,... . i i,k 1 2 i,k 1 2 k n−k,k 1 1 m + 1 i i m+1−i i=k i=1 ! X X (28) Again, without loss of generality we can consider a1 = b1 = 1, ai = i!hi−1(X) and bi = i!hi−1(Y). So, when stated in terms of symmetric functions, this equality can be proved directly using the standard rules involving the sum of two alphabets:

B (X + Y) = (n−k)! h (k(X + Y)) = (n−k)! h (kX)h (kY) n−k,k k! n−2k k! i1+i2=n i1−k i2−k n −1 n (29) = Bi ,k(X)Bi ,k(Y). k i1+i2=n i1 1 2 P P

9 3.4.2 Multiplication by a constant Iterating formula (29), the multiplication of the alphabet by a constant m ∈ N gives

m (n − k)! n −1 n B (mX) = h (k(mX)) = B (X). (30) n−k,k k! n−αk k i , . . . , i ij ,k i i n 1 m j 1+···X+ m= Y=1 Multiplication by a complex constant is related to Jack symmetric polynomials. These polynomials were defined by Henry Jack in 1969 in order to interpolate between Schur functions (α = 1) and zonal polynomials (α = 2) [19, 20]. To be more precise, the Jack polynomials defined up to a normalization coefficient as the unique basis of symmetric functions orthogonal with respect to a one-parameter deformation of the standard scalar product on symmetric functions and which are orthogonal in the monomial basis with respect to the dominance order (see e.g. [23] for more details). β 1 X For partitions with one part, the Jack symmetric function is proportional to gn = hn( β ). (see e.g. [23]). From (16), we then have

1 n! k n! ( β ) B ( X) = h ( X) = g k (X). (31) n,k β k! n−k β k! n−k

Example 3.4 Let us illustrate this property by giving an expression of some rectangular Jack polynomials as a hyperdeterminant of Bell polynomials. First we recall the deﬁnition of a hyperdeterminant: a few years after Cayley introduced the modern notation for determinants [7], he proposed several extensions to higher dimensional arrays under the same name hyperdeterminant [8, 9]. Consider a hypermatrix M = (Mi1,...,ik )1≤i1,...,ik≤n, the hyperdeterminant of M is the polynomial deﬁned by

n 1 Det[k](M) = sign(σ ) . . . sign(σ ) M . n! 1 k σ1(i)...σk(i) σ σ ,...,σ S i = 1 Xk∈ n Y=1 We consider the Jack polynomials with the normalization:

′ α α(λi − j + 1) + λj − i Qλ = ′ mλ + (∗)mµ, α(λi − j) + λj − i + 1 (i,jY)∈λ µX≺λ where mλ is a monomial function (see e.g.[23]). We have:

1 (mn)! m! ( m ) X [2m] X QLn ( ) = n Det ( BL+i1+···+im−im+1···−i2m+m,m( )) n!(m!) (L + i1 + ··· + im − im+1 · · · − i2m + m)!

This is a direct consequence of (31) and the equality

1 n 1 [2m] ( m ) n!(m!) ( m ) Det g (X) = Q n (X) L+i1+i2+···+im−im+1···−i2m (mn)! L which is proved in [24, 2].

10 Example 3.5 A second example of application allows us to slightly extend the deﬁnition of Bell polynomials. ( 1 ) 1 N X n! kα X n! (kα)! X Note that for any α such that α = β ∈ , we have Bn,k(α ) = k! gn−k ( ) = k! (n+kα−k)! Bn+kα−k,kα( ). Mimicking this equality, we deﬁne Bγ,α when γ − kα ∈ N: Γ(γ + 1) k! B (X) := B (αX). (32) γ,kα Γ(kα + 1) (γ − kα + k)! γ−kα+k,k

3.4.3 Product of two alphabets

Let (an)n and (bn)n be two sequences of numbers such that a1 = b1 = 1 and a−n = b−n = 0 for each n ∈ N. Consider also three integers k = k1k2. The following identity seems laborious to prove:

aλi−i+j+1 bλi−i+j+1 Bn,k . . . , n! det det ,... = (λi − i + j + 1)! (λi − i + j + 1)! λ⊢n−1 ! X n! ℓ(λ) Bλi−i+j+k1,k1 (a1 , a2,... ) Bλi−i+j +k2,k2 (b1, b2,... ) (k1!k2!) det det . k! (λi − i + j + k1)! (λi − i + j + k2)! λ⊢n−k X

But it looks rather simpler when we recognize n! B (XY) = h (kXY) (33) n,k k! n−k XY X Y and apply hn(k ) = λ⊢n sλ(k1 )sλ(k2 ). maj(σ)+maj(σ−1 ) For instance, consider the polynomial fn(q) = q where maj(σ) denotes the P σ∈Sn major index of the permutation σ, that is the number of i satisfying σi > σi+1. The generating function 1 P qn n n−1 i of this polynomial is known as i i = 2 fn(q)t , where (α)n = (1 − q α) is the i≥1 (1−q t) n≥0 (q)n i=0 q-Pochhammer symbol (see e.g. [31, Exercise 4.20]). Q P Q n 2 i i n 1 1 1 m P 1+···+ m= h i1,...,im iq Remark that i i = i j = σ . Since h = i≥1 (1−q t) i,j≥1 1−q + t t 1−q n 1−q (q)n n n [n]q! 1−q where Q = Qis the q-binomial and [n]q! = is the q-factorial, we obtain: i1,...,im q [i1]q!...[im]q! 1−q h i 2 λi − i + j i1, . . . , ik n−1 n! i +···+i =λ −i+j q q 1 Xk i det = Bn,k . . . , n! 2 fn−1(q),... . k! (q)λ −i+j (q) λ⊢n−k i n−1 X

4 Bell polynomials and the Faà di Bruno algebra

The algebra of symmetric functions can be endowed with another coproduct that confers a structure of Hopf algebra: this is the Faà di Bruno algebra. This algebra is rather important since it is related to the Lagrange-Bürmann formula. The Bell polynomials appear also in this context. As a consequence, one can deﬁne a new operation on alphabets corresponding to the composition of Cauchy generating functions. We show also that the antipode of the Faà di Bruno algebra can be written in terms of Bell polynomials.

11 4.1 The Arbogast-Faà di Bruno formula The aim of this section is to give a brief account on the relations between the Faà di Bruno formula (in fact certainly due to Louis François Antoine Arbogast in 1800 [1, p. 60]) and the Bell polynomials. This (n) dn formula allows to write the nth derivative of a composition of functions f ◦g in terms of f (t) = dtn f(t) (n) dn and g (t) = dtn g(t). It reads

k dn n! g(λj )(t) f(g(t)) = f (k)(g(t)) . (34) dtn z (λ − 1)! > λ j j Xk 0 λ=(λ1X,...,λk)⊢n Y=1

For our purpose, we will consider f and g as series with g(0) = 0. So we set f(t) = σt(X) and Y X Y Yˆ g(n)(t) n g(t) = tσt( ) where and are two virtual alphabets. Set σ1( (t)) = exp{ n≥1 n! t } (in other Yˆ g(n)(t) n words, pn( (t)) = (n−1)! t ). We recognize from (17): P dn f(g(t)) = f (k)(g(t))B (g′(t), g′′(t),... ) = f (k)(g(t))Bp (Yˆ(t)). dtn n,k n,k k> k> X0 X0 Yt g(n+1)(t) Setting hn( ) = (n+1)!g′(t) , we obtain the equivalent expression dn f(g(t)) = (g′(t))kf (k)(g(t))B (Yt) (35) dtn n,k k> X0 We deﬁne a new operation on alphabets:

σt(X♦Y) := (σt(X) ◦ tσt(Y)). (36)

Straightforwardly, one has n k! h (X♦Y) = h (X)B (Y). (37) n n! k n,k Xk=1 Note that (35) is not more general than (37). Indeed, consider the series (f ◦ g)(x + t) = n≥0(f ◦ (n) xn Xt 1 ′ n f (n)◦g(t) Xt 1 g) (t) n! and the virtual alphabet hn( ) := f◦g(t) (g (t)) n! . One has σx( ) = f◦g(tP) f(g(t) + ′ Yt 1 Xt♦Yt 1 xg (t)) and σx( ) = xg′(t) (g(x + t) − g(t)). Hence, σx( ) = f◦g(t) f ◦ g(x + t). And we recover (35) from (37).

4.2 Faà di Bruno Algebra The operation ♦ does not deﬁne a coproduct which is compatible with the classical product in Sym. In order to construct a combinatorial Hopf algebra, we need to slightly modify the deﬁnition of the composition of alphabets. Let f(t) = tσt(X) and g(t) = tσt(Y). The Faà di Bruno composition is given by: 1 σ (X ◦ Y) = f ◦ g(t) = σ (Y)σ (X♦Y). (38) t t t t

This operation deﬁnes the coproduct given by ∆1(σt) = σt(X ◦ Y), where hλ(X)hµ(Y) is identiﬁed with hλ ⊗ hµ for all partitions λ and µ. Together with the product in Sym, we obtain a Hopf algebra

12 F known as the Faà di Bruno algebra. The relationship with Bell polynomials can be established by observing that, from (38), we have: 1 1 1 f ◦ g(t) = tσ (Y) h (X)(tσ (Y))k = h (X)(tσ (Y))k t t t k t t k−1 t k≥0 k≥1 n+1X X k! = h (X)B (Y) tn, (n + 1)! k−1 n+1,k n ! X≥0 Xk=1 so that n (k + 1)! h (X ◦ Y) = h (X)B (Y) (39) n (n + 1)! k n+1,k+1 k X=0 The algebra F has a structure of Hopf algebra. Since the coproduct is deﬁned in terms of alphabets, it is also the case for the antipode deﬁned as the operation which associates to each alphabet X an h−1i h−1i alphabet X satisfying σt(X ◦ X ) = 1. The antipode of F is explicitly given by the following formula (see e.g. [23, Ex. 24, p. 35 and 25, p. 132]):

h (−(n + 1)X) n! h (Xh−1i) = n = B (−X). (40) n n + 1 (2n + 1)!(n + 1) 2n+1,n

4.3 Lagrange-Bürmann formula In this section, we point out the connexions between the composition of alphabet ◦ (and its associated antipode) and the Lagrange inversion.For our purpose, set ω(t) = tσt(X). We want to ﬁnd an alphabet h−1i Y such that φ(t) = σt(Y) satisﬁes ω(t) = tφ(tω(t)). We have φ(t) = φ(tσt(X ◦ X )) = φ(tσt(X)) ◦ Xh−1i 1 Y Xh−1i σt( ) = h−1i . It follows that = − . σt(X ) Now remark the following fact:

1 [tk] σ (A) − t (σ (A))′ = δ (41) t k t 0,k A A B A 1 A ′ B B ′ B for any alphabet . Setting = k , we have σt( ) − t k (σt( )) = σt(k ) − t(σt( )) σt((k − 1) ). If n and m are two integers and k = n − m, then setting Y = −B, we obtain

m−1 m ′ m ′ t σt(kB) − t (σt(B)) σt((k − 1)B) = (t σt(mY)) σt(−nY). (42)

So, from (41) and (42), one gets

dn−1 tn (umσ (mY))′σ (−nY) = tm. (43) dun−1 u u |u=0 n! n≥1 X h−1i m h−1i m Setting Y = X and ω(t) = tσt(X), the series Fm(t) := t σ(mX ) satisﬁes Fm(ω(t)) = t . Hence from (43), one obtains

m dn−1 n Xh−1i ′ Xh−1i tn Fm(ω(t)) = t = n≥0 dun−1 (u σu(m )) σu(−n ) |u=0 n! dn−1 ′ n tn = PFm(0) + n≥0 dun−1 [Fm(u)(φ(u)) ]|u=0 n! . P 13 Hence, by linearity we recover the classical Lagrange-Bürmann formula for any formal power series F : dn−1 tn F (ω(t)) = F (0) + F ′(u)(φ(u))m . (44) dun−1 |u=0 n! n≥0 X Remark that if we suppose F (t) = σt(X) and ω(t) = tσt(Y), then (44) is equivalent to dn−1 tn σ (X♦Y) = 1 + [σ′ (X)σ (−nYh−1i)] . (45) t dun−1 u u |u=0 n! n X≥1 In other words, n 1 1 h (X♦Y) = (i + 1)h (X)h (−nYh−1i) = kh (X)h (−nYh−1i). (46) n n i+1 j n k n−k i j n k +X= −1 X=1 Considering (46) as a generating function and comparing it with(37), we obtain Yh−1i (k−1)! Y Proposition 4.1 hn−k(−n ) = (n−1)! Bn,k( ). And, as a consequence, Proposition 4.1 and (40) allow us to recover a result due to Sadek Bouroubi X X (n−1)! X and Moncef Abbas [5]: Bn,k(1, h1(2 ), . . . , m!hm((m + 1) ),... ) = (k−1)! hn−k(n ). If we suppose F (t) = tσt(X) (this implies F (0) = 0), then we obtain the following formula which is very close to X Y dn ′ X Yh−1i tn (46): σt( ◦ ) = n≥0 dun [σu( )σu(−(n + 1) )]|u=0 (n+1)! . P 4.4 Bell polynomials of compositions of alphabets From the Cauchy series (36), we observe that k(X♦Y) = (kX)♦Y. The following results give formulas involving Bell polynomials and composition of alphabets. Proposition 4.2 We have: n−k n −1 i + k −1 1. B (X♦Y) = B (X)B (Y), k n,k i i+k,k n−k,i i X=1 n−k n + k n + k 2. B (X ◦ Y) = B (X♦Y)B (Y). n n,k i + k i+k,k n−i,k i X=0 X♦Y n! X♦Y n! X ♦Y Proof First, write Bn,k( ) = k! hn−k(k( )) = k! hn−k((k ) ). Hence, expand the right hand X♦Y n! n−k i! X Y X side as Bn,k( ) = k! i=1 (n−k)! hi(k )Bn,i( ) and replace hi(k ) by its expression as a Bell polynomial. So we obtain the first identity. P The second identity is obtained by remarking that X◦Y = X♦Y+Y and hence k(X◦Y) = k(X♦Y)+kY. X Y n! X♦Y Y n! n−k X♦Y Y So we have Bn,k( ◦ ) = k! hn−k(k( ) + k ) = k! i=0 hi(k( ))hn−k−i(k ) and we recover the identity from (37). P X Y n−k i n j+k −1 X Y Y Combining the two identities, we find Bn,k( ◦ ) = i=0 j=1 i j Bj+k,k( )Bi,j( )Bn−i,k( ). P P 1 n n−1 n Example 4.3 Consider the numbers βn = (n+1)! k=0 k−1 k k!. The integer (n + 1)!βn counts the number of stalactic classes for endofunctions on n letters [18]. The corresponding generating function P (exp{ t }−1)(1−t) n 1−t X♦Y X exp(t)−1 Y 1 is n βnt = t = σt( ) with σt( ) = t and σt( ) = 1−t . So we obtain: n−k 1 n−k−1 Bn,kP(β1, β2,... ) = n! i=1 (i+k)! i−1 Si+k,k. P 14 4.5 A deformation of the Faà di Bruno algebra In [16], Foissy has introduced, as a byproduct of his investigations on the Dyson-Schwinger equations in the Connes-Kreimer algebra, a one-parameter familly ∆γ of coproducts on the algebra of symmetric functions. This family interpolates between the coproduct of the Hopf algebra of symmetric functions (which corresponds to γ = 0), and the coproduct of Faà di Bruno (which corresponds to γ = 1). For R X Y n X Y any γ ∈ , ∆γ is defined by ∆γ(hn) = hn( ◦γ ) := k=0 hk( )hn−k((kγ + 1) ), where for any two integer partitions λ and µ, hλ(X)hµ(Y) is identified with hλ ⊗ hµ. P We can express this coproduct in terms of partial Bell polynomials by

n n 1 kγ + 1 h (X ◦ Y) = h (X)h ((kγ + 1)Y) = k!h (X)B Y . n γ k n−k n! k n,k k Xk=0 Xk=0 Alternatively, with the notations of (32), we obtain:

n Γ n + kγ+1 + 1 X Y k X Y hn( ◦γ ) = hk( )Bn+ kγ+1 −k, kγ+1 ( ). Γ kγ+1 + 1 k k Xk=0 k 5 Word Bell polynomials

The Bell polynomials are related to set partitions. Hence, it is natural that many identities can be understood when stated in a Hopf algebra whose bases are indexed by set partitions. The algebra WSym of word Bell polynomials seems to be a good candidate. We show that most of the identities involving Bell polynomials in Sym can be interpreted in WSym(A) and in fact are consequences of equalities in WSym(A).

5.1 Deﬁnition A Consider a family (Xn)n∈N of homogeneous polynomials Xn ∈ WSymn( ). The word Bell polynomial Bn,k will be a homogeneous polynomial of degree n deﬁned by the generating function

k n 1 m B (X ,...,Xm,... )t = Xmt . (47) n,k 1 k!   n m X X≥1   Note that in (47) ∗ k means ∗ k⊗. (the formal variable t multiplies as tntm = tn+m 6= tn tm). For {{1,...,n}} instance, setting Xn = S (A), we obtain

{{1}} {{1,...,m}} π Bn,k(S (A),...,S (A),... ) = S (A). (48) #π=k Xπn

{{1,...,n}} Another interesting equality is related to Xn = Φ :

{1} {1,...,m} π {{1,...,n}} Bn,k(Φ ,..., (m − 1)!Φ ,... ) = #(π1 − 1)! ··· #(πk − 1)!Φ (A) = S (A). k k #π=k X X Xπn (49)

15 In fact, each polynomial Xn being homogeneous of degree n, we do not need the formal variable t to deﬁne the word Bell polynomials:

k 1 Bn,k(X ,...,Xn,... ) = Xm . (50) 1 k!   n m X X≥1   Nevertheless, t will be useful when we project the identity to non-graded spaces.

Remark that the shuﬄe product splits into two half-products ≺ and ≻ deﬁned by au ≺ bv = a(u bv) and au ≻ bv = b(au v). Remark that this endows ChAi with a structure of Zinbiel algebra. A Zinbiel algebra is a special case of a dendriform algebra, that is a double algebra with two products ≺ and ≻ satisfying 1. (u ≺ v) ≺ w = u ≺ (v ≺ w) + u ≺ (v ≻ w) 2. (u ≻ v) ≺ w = u ≻ (v ≺ w) 3. (u ≺ v) ≻ w + (u ≻ v) ≻ w = u ≻ (v ≻ w) together with an additional equality u ≺ v = v ≻ u. Noting that WSym(A) is stable by ≺ and ≻, this endows WSym(A) with a structure of Zinbiel algebra. With such a structure, (47) can be written as

→ ≺ k

Bn,k(X ,...,Xn,... ) = Xm , (51) 1   n m X X≥1   → → → where x≺ k = x ≺ (x ≺ (x ≺ · · · ≺ (x ≺ x) ··· ). Indeed, we have x≺ k = a(x′ x≺ k − 1) if → ′ A∗ ≺ k 1 ′ k−1 1 k x = ax ∈ . Reasoning by induction on k, we ﬁnd x = (k−1)! a(x x ) = k! x and prove (51).

5.2 Word Bell polynomials and sums of alphabets Let A and B be two (noncommutative) alphabets. For our purpose, we need to introduce the partial shuffle operators, defined in [6], which are multilinear operators Π indexed by set compositions. Let ∗ Π = [π1, . . . , πk] be a set composition and w1, . . . , wk ∈ (A+B) . The partial shuffle Π(w1, . . . , wk) of w1, . . . , wk with respect to Π equals the only word a1 ··· an satisfying aj1 ··· ajℓ = wi if πi = {j1, . . . , jℓ} for each 1 ≤ i ≤ k, when |wi| = #πi for any 1 ≤ i ≤ n and 0 otherwise. We define

{π1,...,πk} SB (A) = ×k   {1} B {1} B {1,··· ,n1−1} A {1,··· ,nk −1} A [{i1},...,{ik },π1\{i1},...,πk\{ik}] zS ( ),...,S}| ( {),S ( ),...,S ( ) . X   i1∈π1,...,ik∈πk   For instance, one has {{1,...,n}} {{1}} {{1,...,n−1}} SB (A) = S (B) S (A). (52)

16 π Let WB(A) be the subalgebra of (ChA ∪ Bi, ) generated by the SB (A). The linear map sending each π π S (A) to SB (A) is an algebra morphism from (WSym(A), ) to WB(A) and we have

k {{1}} A {{1,...,m}} A n 1 {{1,...,m}} A m n Bn,k(SB ( ),..., SB ( ),... )t = k! m≥1 SB ( )t k {{1},...,{k}} B w A P = t SP ( ) σt (k ).

{{1}} {{1,...,m}} {{1},...,{k}} {{1,...,n−k}} So we obtain Bn,k(SB (A),..., SB (A),... ) = S (B) S (kA). Together with (52), this gives an analogue of (16). Proposition 5.1

{{1}} {{1}} {{1,...,m−1}} {{1},...,{k}} {{1,...,n−k}} Bn,k(S (B),..., S (B) S (A),... ) = S (B) S (kA). (53)

B A {{1}} B {{1}} B {{1,...,m−1}} A For simplicity, write Bn,k( ) := Bn,k(S ( ),..., S ( ) S ( ),... ). B Let k = k + k . From S{{1}( ),...,{k1}} S{{1},...,{k2}}(B) = k S{{1},...,{k}}(B), (53) gives 1 2 k1

B k B (A) = k S{{1},...,{k}}(B) S{{1,...,n−k}}(kA) = k1 n,k k1 n−k {{1},...,{k1}} {{1},...,{k2}} {{1,...,i}} {{1,...,n−k−i}} S (B) S (B) S (k1A) S (k2A) = i=0 ! n−k X {{1},...,{k1}} {{1,...,i}} {{1},...,{k2}} {{1,...,n−k−i}} S (B) S (k1A) S (B) S (k2A) i X=0 We deduce an analogue of (22):

Corollary 5.2 Let A and B be two alphabets and k = k1 + k2 be three nonnegative integers. We have

n k B B B B (A) = B (A) B (A). (54) k n,k i,k1 n−i,k2 1 i X=0 A A A {{1,...,n}} A n {{1,...,i}} A {{1,...,n−i}} A Suppose now = 1 + 2. By S ( ) = i=0 S ( 1) S ( 2), (53) gives a word analogue of the convolution formula for Bell polynomials (28). Indeed, we obtain P {{1},...,{k}} B B A {{1},...,{k}} B 2 A S ( ) Bn,k( ) = S ( ) Sn−k,k( ) 2 = S{{1},...,{k}}(B) n−k {{1,...,i}} {{1,...,n−i−k}} S (kA1) S (kA2) i=0 ! n−Xk B A B A = Bi+k,k( 1) Bn−i,k( 2). i X=0 Simplifying, we get: Corollary 5.3 n {{1},...,{k}} B B A B A B A S ( ) Bn,k( ) = Bi,k( 1) Bn−i,k( 2). (55) i X=0

17 5.3 Commutative image Consider the natural projection c : ChA+Bi → C[A+B], that is a morphism of algebra. The restriction π λπ π λπ to WSym(A) sends Φ (A) to p (A) and S = λπ!h (A). Note that if P and Q are two homogeneous polynomials of respective degree p and q, c(P Q) = p+q p c(p)c(q). So k B 1 c B (A)tm = c S{{1}}(B) S{{1,...,m−1}}tn m,k k! m !  m !  X X {{1}...{k}} π #πi+k = c S (B) S (A)tP  π k #X≤

k m λπ m = h (B) λπ!h (A) t . 1 k   m πm k, π k X −X# ≤ Hence,  

n B m n n k λπ m [t ]c B (A)t = [t ]h (B) λπ!h (A) t m,k k 1   m ! m X X π m−Xk, #π≤k n  1 m −k = h (B)k[tn] λ!hλ(A) tm k 1  λ! λ , . . . , λ  m 1 k X λ⊢m−Xk, ℓ(λ)≤k n  (m − k)!  = h (B)k[tn] hλ(A) tm k 1 λ! m   X λ⊢m−Xk, ℓ(λ)≤k n  1  = (n − k)!h (B)k[tn] hλ(A) tm k 1 λ! m   X λ⊢m−Xk, ℓ(λ)≤k  k  n 1 m+1 = n![t ] h (B)hm(A)t k!  1  m X≥1 = Bn,k (h1(B), . . . , m!h1(B)hm−1(A),... ) . Finally, we obtain B A B B A c(Bm,k( )) = Bn,k (h1( ), . . . , m!h1( )hm−1( ),... ) . (56) On the other hand, we have k W A A m c(σt (k )) = c Sm( )t )  m !  X   A A m1+···+mk = c Sm1 ( ) ··· Smk ( )t m ,...,m ! 1X k m + ··· + mk 1 A A m1+···+mk = m1!hm1 ( ) ··· mk!hmk ( )t m1, . . . , mk m1,...,mk X m = m !h (A) ··· m !h (A)tm. m , . . . , m 1 m1 k mk m m +···+m =m 1 k X 1 X k 18 Hence, n W A n A A m [t ]c(σt (k )) = n![t ] hm1 ( ) ··· hmk ( )t m m1+···+mk=m n X X = n![t ]σt(kA) = n!hn(kA). It follows that c(Sn(kA) = n!hn(kA)) = hn(kA). (57) From (56) and (57), (53) projects as e B A {{1},...,{k}} B {{1,...,n−k}} A c(Bn,k( )) = c(S ( ) S (k )) B B A n B k˜ A Bn,k (h1( ), . . . , m!h1( )hm−1( ),... ) = k h1( ) hn−k( ).

Specializing h1(B) = 1, we recover (16). In the same way, (54) (resp. (55)) projects on (22) (resp.(29)).

5.4 Composition of word Bell polynomials

Let k1 and k2 be two positive integers.

k1 B (BB (A),..., BB (A),... )tn = 1 BB (A)tm n n,k1 k2,k2 k2+m−1,k2 k1! m≥1 k2+m−1,k2 k1 {{1},...,{k2k1}} B W A P = t SP ( ) σt (k1k2 ). We deduce B B A A {{1},...,{k2k1}} B {{1,··· ,n−k1}} A Bn,k1 (Bk2,k2 ( ),..., Bk2+m−1,k2 ( ),... ) = S ( ) S (k1k2 ). Hence, Proposition 5.4

B (BB (A),..., BB (A),... ) = BB (A). n,k1 k2,k2 k2+m−1,k2 n−k1+k1k2,k1k2 (58)

A family of functions (fk(a))k is said to be W -binomial if it satisﬁes

fn(a + b) = fi(a) fj(b). (59) n i j X= + We have k n 1 n m B (f (a), . . . , fm (a),... )t = [t ] fm (a)t n,k 0 −1 k!  −1  m≥1 1 X =  f (a) ··· f (a). k! i1 ik i1+···+Xik=n−k Iterating (59), we obtain 1 Proposition 5.5 B (f (a), . . . , f (a),... ) = f (ka). n,k 0 m−1 k! n−k B A We set fn(k) = k!Bn,k( ) and f0(k) = 1. By (54), the family (fn)n∈N is W -binomial. Hence we obtain an analogue of (23): k , . . . , k B A ,... k k B A . 1!Bn,k1 (1 2!Bm−1,k2 ( ) ) = ( 1 2)!Bn−k1,k1k2 ( ) (60)

19 6 Bell polynomials in other Hopf algebras

The algebra WSym∗ ∼ (WSym(A), ) can be realized as a subalgebra of several combinatorial Hopf algebras. Hence, Bell polynomials can be deﬁned in these algebras and their expansions in terms of some combinatorial bases give rise to new identities. Mimicking Section 5, the projection of these identities allows us to recover classical equalities involving Bell polynomials. In this section, we give a few instances of this construction.

6.1 Bi-word Bell polynomials The bi-indexed word algebra BWSym was defined in [6]. We recall its definition here: the bases of BWSym are indexed by set partitions into lists, which can be constructed from a set partition by ordering each block. For instance, {[1, 2, 3], [4, 5]} and {[3, 1, 2], [5, 4]} are two distinct set partitions into lists of the set {1, 2, 3, 4, 5}. The number of set partitions into lists of an n-element set (or set partitions into lists of size n) is given by Sloane’s sequence A000262 [30]. If Πˆ is a set partition into lists ′ ′ ′ of {1, . . . , n}, we will write Πˆ n. Set Πˆ ⊎Πˆ = Πˆ ∪{[l1 +n, . . . , lk +n]:[l1, . . . , lk] ∈ Πˆ } n+n . Let Πˆ ′ ⊂ Πˆ n. Since the integers appearing in Πˆ ′ are all distinct, the standardized std(Πˆ ′) of Πˆ ′ is well defined as the unique set partition into lists obtained by replacing the ith smallest integer in Πˆ by i. For example, std({[5, 2], [3, 10], [6, 8]}) = {[3, 1], [2, 6], [4, 5]}. The set partitions into lists are partially ordered by the Πˆ ′ ≤ Πˆ if and only if the lists of Πˆ ′ are obtained by breaking the lists of Π. For instance {[3, 1, 4], [2, 6], [5]} ≤ {[3, 1, 4], [5, 2, 6]}, {[3, 1, 4], [2, 5, 6]} but {[3, 1, 4], [2, 6], [5]} 6≤ {[3, 1, 4], [6, 5, 2]}. The Hopf algebra BWSym is formally defined by its basis (ΦΠˆ ) where the Πˆ are set partitions into ′ ′ lists, its product ΦΠˆ ΦΠˆ = ΦΠˆ⊎Πˆ and its coproduct ∧ ˆ ˆ ′ ˆ ′′ ∆(ΦΠ) = Φstd(Π ) ⊗ Φstd(Π ), (61)

∧ X where the means that the sum is over the (Πˆ ′, Πˆ ′′) such that Πˆ ′ ∪ Πˆ ′′ = Πˆ and Πˆ ′ ∩ Πˆ ′′ = ∅. An Πˆ analogue ofX monomial functions is deﬁned by Φ = MΠˆ ′ . By triangularity, ΠˆX≤Πˆ ′ ∧ ˆ ˆ ′ ˆ ′′ ∆(MΠ) = Mstd(Π ) ⊗ Mstd(Π ). (62) Hence, the dual algebra BWSym∗ is generatedX by the dual basis (Φ∗ ) of (Φ ). So, from (61), we obtain Πˆ Πˆ ′′ ′′ Φ∗ Φ∗ = αˆΠˆ Φ∗ . The dual basis (S ) of (M ) satisﬁes S = Φ∗ , where αˆΠˆ denotes Πˆ Πˆ ′ Πˆ ′′ Πˆ ,Πˆ ′ Πˆ ′′ Πˆ Πˆ Πˆ Πˆ ≤Πˆ ′ Πˆ Πˆ ,Πˆ ′ ˆ ′′ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ the numberP of ways to write Π = Π1 ∪ Π2 with std(Π1) = Π and std(PΠ2) = Π and Π1 ∩ Π2 = ∅. And from (62), we have Πˆ ′′ S ˆ S ˆ ′ = αˆ ˆ ˆ ′ S ˆ ′′ . Π Π2 Π,Π Π XΠˆ ′′ Hence, the subalgebra generated by the polynomials Sπ = s(Π)=ˆ π SΠˆ , where π is a set partition and s({[i1, . . . , i1 ], ··· , [ik, . . . , ik ]}) = {{i1, . . . , i1 }, ··· , {ik, . . . , ik }}, is isomorphic to WSym⋆. 1 ℓ1 1 ℓk 1 ℓ1 1P ℓk Indeed

Πˆ ′′ SπS ′ = S S ′ = αˆ S ′′ . π Πˆ Πˆ  Πˆ ,Πˆ ′  Πˆ s(Π)=ˆ π,sX(Πˆ ′)=π′ XΠˆ ′′ s(Π)=ˆ π,sX(Πˆ ′)=π′   20 ′′ Πˆ ′′ s(Πˆ ) π′′ But αˆ = α where α ′ is the number of ways to write the set partition s(Π)=ˆ π,s(Πˆ ′)=π′ Πˆ ,Πˆ ′ s(Π)ˆ ,s(Πˆ ′) π,π ′′ ′ π asP the disjoint union π1 ∪ π2 with std(π1) = π and std(π2) = π . So,

s(Π′′) π′′ π′′ SπSπ′ = απ,π′ SΠˆ ′′ = απ,π′ SΠˆ ′′ . = απ,π′ Sπ′′ . ′′ ′′ XΠˆ ′′ Xπ s(ΠˆX′′)=π′′ Xπ ⋆ ∗ Hence, WSym is isomorphic as an algebra to the subalgebra of BWSym generated by (Sπ) and the ex- π A plicit isomorphism sends Sπ (∼ S ( )) to Sπ. Hence, the polynomial Bn,k S{1}, S{1,2},..., S{1,...,m},... is well deﬁned. So (48) implies

Bn,k S{{1}}, S{{1,2}},..., S{{1,...,m}},... = Sπ = SΠˆ = SΠˆ . π n π n s(Π)ˆ Πˆ n #Xπ=k #Xπ=k X #XΠ=ˆ k σ S σ σ , . . . , σ If for each permutation ∈ n, we set [ ] = {[ 1 n]}, then S{{1,··· ,n}} = σ∈Sn Sσ and the previous formula becomes P

Bn,k S1, S12 + S21,..., Sσ,... = SΠˆ . (63) σ∈Sm ! Πˆ n X #XΠ=ˆ k

This provides a bi-indexed word analogue of the classical formula Bn,k(1!, 2!, ··· , m!, ··· , ) = Ln,k (the Lah number Ln,k counts the number of partitions of the set {1, . . . , n} into k lists).

6.2 Cycle Bell polynomials We consider the Hopf algebra SQSym which is the dual of the Grossman-Larson Hopf algebra of heap-ordered trees [17]. The combinatorics of this algebra have been extensively investigated in [18]. In particular, it is shown that SQSym has a polynomial realization spanned by the Mσ = x x σ S et al. i1<···

{{1}} {{1,2}} {{1,2,...,m}} is clearly the image of Bn,k(S ,S ,...,S ,... ) by this isomorphism. Denoting by Cn the set of the cycles of size n, we obtain

Bn,k(M1,M21,M231 + M312,..., M{{1,2,...,m}},... ) = Mσ. (64) S σ∈Cn σ∈ n X #supp(Xσ)=k

This gives an analogue of the formula Bn,k(0!,..., (m − 1)!,... ) = |sn,k|.

21 6.3 Word symmetric functions of level 2

Π We consider the algebra WSym(2) which is spanned by the Φ where Π is a partition of a partition ⊢ Π Π′ Π∪Π′[n] π of {1, . . . , n} (we will denote Π ⊢ n). The product of this algebra is given by Φ Φ = Φ where Π′[n] = {e[n]: e ∈ Π′}. The dimensions of this algebra are given by the exponential generating i (2) t function i bi i! = exp(exp(exp(t − 1) − 1). The ﬁrst values are P 1, 3, 12, 60, 358, 2471, 19302, 167894, 1606137,... see sequence A000258 of [30]. The partitions of partitions are in bijection with the pairs of partitions

(π1, π2) such that π1 ≤ π2. The bijection is given by pair(Π) = e∈Π e, f∈e f : e ∈ Π . For instance n o pair({{{2, 5}, {1, 4, 6}}, {{3}, {7}}, {{8}}}) = S S ({{2, 5}, {1, 4, 6}, {3}, {7}, {8}}, {{2, 5, 1, 4, 6}, {3, 7}, {8}}) .

pair(Π) Π Π std(Π′) So alternatively, we deﬁne Φ = Φ . The coproduct is deﬁned by ∆(Φ ) = Π′∪Π′′=Π Φ ⊗ Π′∩Π′′=∅ std(Π′′) Φ where, if Π is a partition of a partition of {i1, . . . , ik}, std(Π) denotes theP standardized of Π, that is the partition of partition of {1, . . . , k} obtained by substituting each occurence of ij by j in Π. ∗ The coproduct being co-commutative, the dual algebra WSym(2) is commutative. ∗ Π′′ Π′′ The algebra WSym(2) is spanned by a basis (SΠ)Π satisfying SΠSΠ′ = Π′′ CΠ,Π′ SΠ′′ where CΠ,Π′ is the number of ways to write Π′′ = A ∪ B with A ∩ B = ∅, std(A) = Π and std(B) = Π′. Let S P π = pair(Π)=(π′,π) SΠ. We have P

S S = CΠ3 S =  CΠ3  S . π1 π2 Π1,Π2 Π3 Π1,Π2 Π3 pair(Π )=(π′ ,π ) pair(Π )=(π′ ,π ) 1 1 1 Π3 Π3 1 1 1 X ′ X X  X ′  pair(Π2)=(π ,π2)  pair(Π2)=(π ,π2)  2  2    We remark that CΠ3 = απ3 (65) Π1,Π2 π1,π2 pair(Π )=(π′ ,π ) 1 1 1 Xπ′ ,π pair(Π2)=( 2 2) ′ if pair(Π3) = (π3, π3). Indeed, for a given Π3 and a pair (π1, π2), there exists at most one pair (Π1, Π2) such that CΠ3 6= 0 and pair(Π ) = (π′, π ) for i = 1, 2. Furthermore, if such a pair exists, one has Π1,Π2 i i i CΠ3 = απ3 . So the sum of the right hand sides of (65) reduces to one term which equals απ3 . Π1,Π2 π1,π2 π1,π2 So, we obtain S S π3 π3 S π1 π2 = απ1,π2 SΠ3 = απ1,π2 π3 . π3 Π =(π′ ,π ) π3 X 3 X3 3 X ⋆ Equivalently, the subspace spanned by the Sπ is a subalgebra isomorphic to WSym . Hence, we compute the polynomial

S S S S Bn,k( {{1}}, {{1,2}},..., {{1,...,n}},... ) = π, πn #Xπ=k

22 and we obtain

Bn,k(S{{{1}}},S{{{1,2}}} + S{{{1},{2}}},..., S{π},... ) = SΠ. πn ⊢ X ΠX| ⊢ n #Π=k

(2) The formula projects on Bn,k(b1, b2, . . . , bm,... ) = sn,k, where bm denotes the mth Bell number and (2) (2) sn,k is the number of partitions of partitions of {1, . . . , n} into k sets. The triangle of the numbers sn,k is given by the sequence A039810 [30].

7 Conclusion

In brief, we have shown that most of the identities on Bell polynomials belong to one of the two types: those from identities on symmetric functions and those from identities in the Faà di Bruno algebra. In Section 3, we investigated the ﬁrst kind that involves equalities on combinatorial objects. These results can be reach back to Hopf algebras whose bases are indexed by the combinatorial objects. The prototype and most important example is given by word Bell polynomials (Section 5) and a few other examples are investigated in Section 6. Note that one can ﬁnd several other instances of this phenomenon, considering word symmetric functions of level k, set matrix symmetric functions etc. The second kind of identities are related to a more algebraic topic, the Lagrange inversion via the Faà di Bruno algebra. In a forthcoming paper, we will investigate a word analogue of these constructions that is related to the notion of operad.

Acknowledgment: This paper is partially supported by the PHC MAGHREB project IThèM and the ANR project CARMA.

References

[1] L.F.A. Arbogast, Du calcul des dérivations, LeVrault Frères, Strasbourg, 1800. [2] H. Belbachir, A. Boussicault, J.-G. Luque, Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the Vandermonde, Journal of Algebra 320 (2008) 3911–3925 [3] E. T. Bell, Exponential polynomials, Annals of Mathematics 35, (1934), pp. 258–277. [4] N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Sym- metric Group in Noncommuting Variables, Canad. J. Math. 60, no. 2, 266–296(2008). [5] S. Bouroubi, M. Abbas, New identities for Bell’s polynomials New approaches, Rostock. Math. Kolloq 61: 49–55 (2006). [6] J. Bultel, A. Chouria, J.-G. Luque and O. Mallet, Word symmetric functions and the Redﬁeld- Pólya theorem, DMTCS Proceedings As - 25th International Conference on Formal Power Series and Algebraic Combinatorics, (2013). [7] A. Cayley, On the theory of determinants, Trans. Cambridge Philos. Soc. 8 (1843) 1–16. [8] A. Cayley, Mémoire sur les hyperdéterminants, J. Reine Angew. Math. 30 (1846) 1–37. [9] A. Cayley, On the theory of permutants, Cambridge Dublin Math. J. 7 (1852) 40–51.

23 [10] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Pub- lishing Company, Dordrecht-Holland/Boston-U.S., 1974. Steven Roman, The Umbral Calculus, Dover Publications. [11] Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J., Knuth, D. E. (1996). "On the Lambert W function". Advances in Computational Mathematics (Berlin, New York: Springer-Verlag) 5: 329–359. [12] D.Cvijović, New identities for the partial Bell polynomials, Applied Mathematics Letters, 24 (2011), 1544–1547. [13] G. H. E. Duchamp, J.-G. Luque, J.-C. Novelli, C. Tollu, F. Toumazet, Hopf algebras of diagrams, International Journal of Algebra and Computation 21, 3 (2011) 1–23. [14] F. Faà di Bruno, Sullo sviluppo delle Funzioni, Annali di Scienze Matematiche e Fisiche 6 (1855), pp. 479–480. [15] F. Faà di Bruno, Note sur une nouvelle formule de calcul différentiel, Quarterly J. Pure Appl. Math. 1 (1857) 359–360. [16] L. Foissy, Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson- Schwinger equations, Adv. Math. 218(1): 136–162 (2008). [17] R. Grossman and R. G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1989), n. 1, 184–210. [18] F. Hivert, J.-C. Novelli, J.-Y. Thibon, Commutative Combinatorial Hopf Algebras, J. Algebraic Combin. 28 (2008), no. 1, 65–95. [19] H. Jack,A class of symmetric polynomial with a parameter, Proc. Roy. Soc. Edinburgh Sect. A 69 (1970) 1–18. [20] H. Jack, A surface integral and symmetric functions, Proc. Roy. Soc. Edinburgh Sect. A 69 (1972) 347–363. [21] Khelifa, S., and Cherruault, Y. : New results on Adomian method. Kybernetes. Vol. 29; n3, (2000) 332–354 [22] A. Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, CBMS Regional Conference Series in Mathematics, American Mathematical Society (2003). [23] I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed, Oxford University Press, 1995. [24] S. Matsumoto, Hyperdeterminantal expressions for Jack functions of rectangular shapes, J. Al- gebra 320 (2008) 612–632. [25] M. Mihoubi, Bell polynomials and binomial type sequences, Discrete Mathematics, 308 (2008), pp. 2450–2459. [26] M.Mihoubi, Polynômes multivariés de Bell et polynômes de type binomial, L’Université des Sci- ences et de la Technologie Houari Boumediene (2008). [27] J. Riordan, Derivatives of composite functions, Bull. Amer. Math. Soc. 52 (1946), pp. 664–667. [28] J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, New York, (1958) ; Princeton University Press, Princeton, NJ, (1980).

24 [29] M. Rosas, B. Sagan, Symmetric functions in noncommuting variables, Trans. Amer. Math. Soc. 358(1): 215–232 (2006). [30] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. http://oeis.org/. [31] R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999.