869 research outputs found
Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras
Bell polynomials appear in several combinatorial constructions throughout
mathematics. Perhaps most naturally in the combinatorics of set partitions, but
also when studying compositions of diffeomorphisms on vector spaces and
manifolds, and in the study of cumulants and moments in probability theory. We
construct commutative and noncommutative Bell polynomials and explain how they
give rise to Fa\`a di Bruno Hopf algebras. We use the language of incidence
Hopf algebras, and along the way provide a new description of antipodes in
noncommutative incidence Hopf algebras, involving quasideterminants. We also
discuss M\"obius inversion in certain Hopf algebras built from Bell
polynomials.Comment: 37 pages, final version, to appear in IJA
Noncommutative binomial theorem, shuffle type polynomials and Bell polynomials
In this paper we use the Lyndon-shirshov basis to study the shuffle type
polynomials. We give a free noncommutative binomial (or multinomial) theorem in
terms of the Lyndon-Shirshov basis. Another noncommutative binomial theorem
given by the shuffle type polynomials with respect to an adjoint derivation is
established. As a result, the Bell differential polynomials and the -Bell
differential polynomials can be derived from the second binomial theorem. The
relation between the shuffle type polynomials and the Bell differential
polynomials is established. Finally, we give some applications of the free
noncommutative binomial theorem including application of the shuffle type
polynomials to bialgebras and Hopf algebras.Comment: 25 pages, no figure
Bell polynomials in combinatorial Hopf algebras
We introduce partial -Bell polynomials in three combinatorial Hopf
algebras. We prove a factorization formula for the generating functions which
is a consequence of the Zassenhauss formula.Comment: 7 page
Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables
We analyze the structure of the algebra N of symmetric polynomials in
non-commuting variables in so far as it relates to its commutative counterpart.
Using the "place-action" of the symmetric group, we are able to realize the
latter as the invariant polynomials inside the former. We discover a tensor
product decomposition of N analogous to the classical theorems of Chevalley,
Shephard-Todd on finite reflection groups.Comment: 14 page
Cumulants and convolutions via Abel polynomials
We provide an unifying polynomial expression giving moments in terms of
cumulants, and viceversa, holding in the classical, boolean and free setting.
This is done by using a symbolic treatment of Abel polynomials. As a
by-product, we show that in the free cumulant theory the volume polynomial of
Pitman and Stanley plays the role of the complete Bell exponential polynomial
in the classical theory. Moreover via generalized Abel polynomials we construct
a new class of cumulants, including the classical, boolean and free ones, and
the convolutions linearized by them. Finally, via an umbral Fourier transform,
we state a explicit connection between boolean and free convolution
Bell polynomials in combinatorial Hopf algebras
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{\`a} di Bruno Hopf algebra and the
Lagrange-B{\"u}rmann formula
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