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1. Introduction Multivariate Partial Bell Polynomials (Bell Polynomials for Short) Have Been DeNed by E.T

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1. Introduction Multivariate Partial Bell Polynomials (Bell Polynomials for Short) Have Been DeNed by E.T Séminaire Lotharingien de Combinatoire 75 (2017), Article B75h WORD BELL POLYNOMIALS AMMAR ABOUDy, JEAN-PAUL BULTEL{, ALI CHOURIAz, JEAN-GABRIEL LUQUEz, AND OLIVIER MALLETz Abstract. Multivariate partial Bell polynomials have been dened by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Alge- bra, Probabilities, etc. Many of the formulas on Bell polynomials involve combinatorial objects (set partitions, set partitions into lists, permutations, etc.). So it seems natural to investigate analogous formulas in some combinatorial Hopf algebras with bases indexed by these objects. In this paper we investigate the connections between Bell polynomials and several combinatorial Hopf algebras: the Hopf algebra of symmetric functions, the Faà di Bruno algebra, the Hopf algebra of word symmetric functions, etc. We show that Bell polynomials can be dened in all these algebras, and we give analogs of classical results. To this aim, we construct and study a family of combinatorial Hopf algebras whose bases are indexed by colored set partitions. 1. Introduction Multivariate partial Bell polynomials (Bell polynomials for short) have been dened by E.T. Bell in [1] in 1934. But their name is due to Riordan [29], who studied the Faà di Bruno formula [11, 12] allowing one to write the nth derivative of a composition f ◦ g in terms of the derivatives of f and g [28]. The applications of Bell polynomials in Combinatorics, Analysis, Algebra, Probability Theory, etc. are so numerous that it would take too long to exhaustively list them here. Let us give only a few seminal examples. • The main applications to Probability Theory are based on the fact that the nth moment of a probability distribution is a complete Bell polynomial of the cumu- lants. • Partial Bell polynomials are linked to Lagrange inversion. This follows from the Faà di Bruno formula. • Many combinatorial formulas for 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 [7]. The Ph.D. thesis of Mihoubi [24] contains a rather complete survey of the applications of these polynomials together with numerous formulas. Some of the simplest formulas are related to the enumeration of combinatorial objects (set partitions, set partitions into lists, permutations, etc.). So it seems natural to in- vestigate analogous formulas in some combinatorial Hopf algebras with bases indexed by these objects. We recall that combinatorial Hopf algebras are graded bialgebras with 2010 Mathematics Subject Classication. 57T05, 11B73, 11B75. Key words and phrases. Bell polynomials, Munthe-Kaas polynomials, set partitions, colored set par- titions, Hopf algebras, symmetric functions, word symmetric functions, Faà di Bruno algebra, Lagrange inversion. 2 A. ABOUD, J.-P. BULTEL, A. CHOURIA, J.-G. LUQUE, AND O. MALLET bases indexed by combinatorial objects such that the product and the coproduct have some compatibilities. This paper is organized as follows. In Section2, we investigate the combinatorial prop- erties of colored set partitions. Section3 is devoted to the study of the Hopf algebras of colored set partitions. After having introduced this family of algebras, we give some special cases which can be found in the literature. The main application explains the con- nections with Sym, the algebra of symmetric functions. This explains that we can recover some identities for Bell polynomials when the variables are specialized to combinatorial numbers from analogous identities in some combinatorial Hopf algebras. We show that the algebra WSym of word symmetric functions has an important role for this construction. In Section4, we give a few analogs of complete and partial Bell polynomials in WSym, ∗, and where is an innite alphabet and ΠQSym = WSym ChAi A = fa1;:::; an;::: g investigate their main properties. Finally, in Section5 we investigate the connection with other noncommutative analogs of Bell polynomials dened by Munthe-Kaas [33]. 2. Definition, background and basic properties of colored set partitions 2.1. Colored set partitions. Let a = (am)m≥1 be a sequence of nonnegative integers. A colored set partition associated with the sequence a is a set of pairs Π = f[π1; i1]; [π2; i2];:::; [πk; ik]g such that is a partition of for some , and π = fπ1; : : : ; πkg f1; : : : ; ng n 2 N 1 ≤ i` ≤ a#π` for 1 ≤ ` ≤ k, where #s denotes the cardinality of the set s. The integer n is the size of Π. We write jΠj = n, Π n, and Π V π. We denote the set of colored partitions of size n associated with the sequence a by CPn(a). Notice that these sets are nite. We also set S . We endow with the additional statistic , and set CP(a) = n CPn(a) CP #Π CPn;k(a) = fΠ 2 CPn(a) : #Π = kg: Example 1. Consider the sequence whose rst terms are a = (1; 2; 3;::: ). The colored partitions of size 3 associated with a are CP3(a) = ff[f1; 2; 3g; 1]g; f[f1; 2; 3g; 2]g; f[f1; 2; 3g; 3]g; f[f1; 2g; 1]; [f3g; 1]g; f[f1; 2g; 2]; [f3g; 1]g; f[f1; 3g; 1]; [f2g; 1]g; f[f1; 3g; 2]; [f2g; 1]g; f[f2; 3g; 1]; [f1g; 1]g; f[f2; 3g; 2]; [f1g; 1]g; f[f1g; 1]; [f2g; 1]; [f3g; 1]gg: The colored partitions of size 3 and cardinality 2 are CP3;2(a) = ff[f1; 2g; 1]; [f3g; 1]g; f[f1; 2g; 2]; [f3g; 1]g; f[f1; 3g; 1]; [f2g; 1]g; f[f1; 3g; 2]; [f2g; 1]g; f[f2; 3g; 1]; [f1g; 1]g; f[f2; 3g; 2]; [f1g; 1]gg: It is well-known (see, e.g., [24]) that the number of colored set partitions of size n for a given sequence a = (an)n is equal to the evaluation of the complete Bell poly- nomial An(a1; : : : ; am;::: ). It is also known that the number of colored set partitions of size n and cardinality k is given by the evaluation of the partial Bell polynomial Bn;k(a1; a2; : : : ; am;::: ). That is, #CPn(a) = An(a1; a2;::: ) and #CPn;k(a) = Bn;k(a1; a2;::: ): Now, let Π = f[π1; i1];::: [πk; ik]g be a set such that the πj's are nite sets of nonnegative integers with the property that no integer belongs to more than one , and πj 1 ≤ ij ≤ a#(πj ) WORD BELL POLYNOMIALS 3 for any j. Then the standardization std(Π) of Π is well-dened as the unique colored set partition obtained by replacing the ith smallest integer in the πj's by i. Example 2. For instance, we have std(f[f1; 4; 7g; 1]; [f3; 8g; 1]; [f5g; 3]g) = f[f1; 3; 5g; 1]; [f2; 6g; 1]; [f4g; 3]g : We dene two binary operations, ] : CPn;k(a) ⊗ CPn0;k0 (a) −! CPn+n0;k+k0 (a), Π ] Π0 = Π [ Π0[n]; where Π0[n] means that we add n to each integer occurring in the sets of Π0, and : , d CPn;k ⊗ CPn0;k0 −! P(CPn+n0;k+k0 ) 0 ^ ^ 0 ^ and ^ 0 0 Π d Π = fΠ [ Π 2 CPn+n0;k+k0 (a) : std(Π) = Π std(Π ) = Π g: Example 3. We have f[f1; 3g; 5]; [f2g; 3]g ] f[f1g; 2]; [f2; 3g; 4]g = f[f1; 3g; 5]; [f2g; 3]; [f4g; 2]; [f5; 6g; 4]g; and f[f1g; 5]; [f2g; 3]g d f[f1; 2g; 2]g = ff[f1g; 5]; [f2g; 3]; [f3; 4g; 2]g; f[f1g; 5]; [f3g; 3]; [f2; 4g; 2]g; f[f1g; 5]; [f4g; 3]; [f2; 3g; 2]g; f[f2g; 5]; [f3g; 3]; [f1; 4g; 2]g; f[f2g; 5]; [f4g; 3]; [f1; 3g; 2]g; f[f3g; 5]; [f4g; 3]; [f1; 2g; 2]gg: The operator d provides an algorithm which computes all colored partitions: ai1 aik [ [ [ (2.1) CPn;k(a) = ··· f[f1; : : : ; i1g; j1]g d ··· d f[f1; : : : ; ikg; jk]g: i1+···+ik=n j1=1 jk=1 Nevertheless, some colored partitions are generated more than once using this process. For a triple 0 00 , we denote by Π the number of pairs of disjoint subsets ^ 0, (Π; Π ; Π ) αΠ0;Π00 (Π Π^ 00) of Π such that Π^ 0 [ Π^ 00 = Π, std(Π^ 0) = Π0, and std(Π^ 00) = Π00. Remark 4. Notice that, for a = 1 = (1; 1;::: ) (i.e., the ordinary set partitions), there is an alternative way to construct the set CPn(1) eciently. It suces to use the induction CP0(1) = f;g; CPn+1(1) = fπ [ ffn + 1gg : π 2 CPn(1)g [ f(π n feg) (2.2) [ fe [ fn + 1gg : π 2 CPn(1); e 2 πgg: By the application of this recurrence, the set partitions of CPn+1(1) are each obtained exactly once from the set partitions of CPn(1). 2.2. Generating functions. The generating functions of the colored set partitions CP(a) is obtained from the cycle generating function for the species of colored set partitions. The construction is rather classical, see, e.g., [3]. Recall rst that a species of structures is a rule F which produces for each nite set U, a nite set F [U], and for each bijection φ : U −! V , a function F [φ]: F [U] −! F [V ] satisfying the following properties: • for all pairs of bijections φ : U −! V and : V −! W , we have F [ ◦ φ] = F [ ] ◦ F [φ]; • if IdU denotes the identity map on U, then F [IdU ] = IdF [U]. 4 A. ABOUD, J.-P. BULTEL, A. CHOURIA, J.-G. LUQUE, AND O. MALLET An element s 2 F [U] is called an F -structure on U.
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