[Paper Review] Bell polynomials in combinatorial Hopf algebras

Ammar Aboud, Jean-Paul Bultel+3 more|arXiv (Cornell University)|Feb 12, 2014

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper establishes a deep connection between partial and complete Bell polynomials and combinatorial Hopf algebras by introducing colored set partitions as a unifying framework. It constructs a family of Hopf algebras based on these partitions and demonstrates that Bell polynomials can be naturally defined within algebras such as symmetric functions, word symmetric functions (WSym), and the dual of the partition quasisymmetric functions (ΠQSym), recovering classical identities through algebraic structures like dendriform and Zinbiel operations.

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 formulae on Bell polynomials involve combinatorial objects (set partitions, set partitions in lists, permutations, etc.). So it seems natural to investigate analogous formulae 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 formulae). We investigate also the connexion with the Fa{à} di Bruno Hopf algebra and the Lagrange-B{ü}rmann formula.

Motivation & Objective

• To generalize classical Bell polynomial identities to the setting of combinatorial Hopf algebras with bases indexed by combinatorial objects such as set partitions.
• To define and study a new family of combinatorial Hopf algebras based on colored set partitions, parameterized by a sequence of nonnegative integers.
• To establish that the algebra of word symmetric functions (WSym) plays a central role in realizing analogues of Bell polynomials in noncommutative settings.
• To recover classical identities involving Bell polynomials—such as those related to Lagrange inversion and Stirling numbers—via algebraic structures in Hopf algebras.
• To explore connections between Bell polynomials and noncommutative analogues via Zinbiel and dendriform structures, particularly in ΠQSym and the free algebra ℂ⟨A⟩.

Proposed method

• Introduces colored set partitions as a generalization of set partitions, where each block is assigned a color from a sequence a = (a_m), with a_m specifying the number of available colors for blocks of size m.
• Constructs a family of combinatorial Hopf algebras, denoted H_a, with bases indexed by colored set partitions, and equips them with a product and coproduct compatible with the combinatorial structure.
• Defines analogues of complete and partial Bell polynomials in the algebras WSym, ΠQSym = WSym*, and ℂ⟨A⟩, using the standardization map and generating functions.
• Utilizes the Zinbiel algebra structure on ΠQSym, defined via two nonassociative products ≺ and ≻ satisfying specific axioms, to express generating functions for Bell-type polynomials.
• Derives a quasideterminant-type formula for the generating series of partial Bell polynomials in ΠQSym using the Zinbiel product and upper triangular matrices over the algebra.
• Establishes a connection between the polynomial P(M_n; t) defined via iterated ≺-products and the generating series of Bell polynomials, showing that [t^k]P(M_n; t) = ℬ_{n,k}(Φ_{‗{1}‗}, Φ_{‗{1,2}‗}, ...), where M_n is a matrix of basis elements.

Experimental results

Research questions

• RQ1 How can classical Bell polynomial identities be generalized within the framework of combinatorial Hopf algebras?
• RQ2 What is the role of colored set partitions in unifying the algebraic structures underlying symmetric functions, word symmetric functions, and the Faà di Bruno algebra?
• RQ3 How do Zinbiel and dendriform structures in ΠQSym enable the construction of noncommutative analogues of Bell polynomials?
• RQ4 Can quasideterminant formulas from matrix theory be interpreted as generating functions for Bell polynomials in noncommutative Hopf algebras?
• RQ5 What is the precise algebraic mechanism that allows the recovery of combinatorial identities (e.g., involving Stirling or Lah numbers) from Hopf algebra identities in WSym and ΠQSym?

Key findings

• The number of colored set partitions of size n with k blocks is given by the partial Bell polynomial B_{n,k}(a_1, a_2, ...), and the total number of such partitions is A_n(a_1, a_2, ...), the complete Bell polynomial.
• The generating series for the partial Bell polynomials in the algebra ΠQSym is given by ∑_{n} ℬ_{n,k}(Φ_{‗{1}‗}, Φ_{‗{1,2}‗}, ...) t^n = (∑_i Φ_{‗{1,...,i}‗} t^i )^{≺→k}, where the superscript denotes iterated ≺-product.
• The polynomial P(M_n; t), defined via iterated Zinbiel products on a matrix M_n of basis elements of ΠQSym, satisfies [t^k]P(M_n; t) = ℬ_{n,k}(Φ_{‗{1}‗}, Φ_{‗{1,2}‗}, ...), providing a noncommutative analogue of Bell polynomial generation.
• The quasideterminant formula for the matrix M_n with entries Φ_{‗{1,...,j-i+1}‗} yields the generating function for Bell polynomials, generalizing classical results from Ebrahimi et al. and Gelfand et al.
• The Hopf algebra WSym plays a central role in realizing the algebraic structure of Bell polynomials in the noncommutative setting, with its dual ΠQSym supporting the Zinbiel structure used to define the generating functions.
• The construction recovers classical identities such as those linking Bell polynomials to Lagrange inversion and to the moments and cumulants in probability, now interpreted as algebraic identities in Hopf algebras.