In this paper, by means of the classical Lagrange inversion formula, we establish a general nonlinear inverse relations which is a partial solution to the problem proposed in the paper [J. Wang, Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula, J. Integer Seq., Vol. 22 (2019), Article 19.3.8. (https://cs.uwaterloo.ca/journals/JIS/VOL22/Wang/wang53.pdf). As applications of this inverse relation, we not only find a short proof of another nonlinear inverse relation due to Birmajer et al., but also set up a few convolution identities concerning the Mina polynomials.
We give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.
We compute the noncommutative Frobenius characteristic of the natural action of the 0-Hecke algebra on parking functions, and obtain as corollaries various forms of the noncommutative Lagrange inversion formula.
In a previous paper, we showed that $3\bar{5}241$-avoiding permutations are counted by the unique sequence that starts with a 1 and shifts left under the self-composition transform. The proof uses a complicated bijection. Here we give a much simpler proof based on Lagrange inversion.
We introduce a new pair of mutually dual bases of noncommutative symmetric functions and quasi-symmetric functions, and use it to derive generalizations of several results on the reduced incidence algebra of the lattice of noncrossing partitions. As a consequence, we obtain a quasi-symmetric version of the Farahat-Higman algebra.
We consider formal power series defined through the functional q-equation of the q-Lagrange inversion. Under some assumptions, we obtain the asymptotic behavior of the coefficients of these power series. As a by-product, we show that, via the 1/q-Borel transform, the q-Lagrange inversion formula provides an interpolation between the usual Lagrange inversion (q=1) and the probabilistic theory of renewal sequences (q tends to 0). We also discuss some new solutions of the q-Lagrange inversion equation which do not vanish at 0.
We provide operadic interpretations for two Hopf subalgebras of the algebra of parking functions. The Catalan subalgebra is identified with the free duplicial algebra on one generator, and the Schröder subalgebra is interpreted by means of a new operad, which we call triduplicial. The noncommutative Lagrange inversion formula is then interpreted in terms of duplicial structures. The generic solution of the noncommutative inversion problem appears as the formal sum of all parking functions. This suggests that combinatorial generating functions derived by functional inversion should be obtainabl…
We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.
We define, for a somewhat standard forgetful functor from nonsymmetric operads to weight graded associative algebras, two functorial "enveloping operad" functors, the right inverse and the left adjoint of the forgetful functor. Those functors turn out to be related by operadic Koszul duality, and that relationship can be utilised to provide examples showing limitations of two standard tools of the Koszul duality theory. We also apply these functors to get a homotopical algebra proof of the Lagrange inversion formula.
This paper is devoted to the computations of some relevant quantities associated with the free unitary Brownian motion. Using the Lagrange inversion formula, we first derive an explicit expression for its alternating star cumulants of even lengths and relate them to those having odd lengths by means of a summation formula for the free cumulants with product as entries. Next, we use again this formula together with a generating series for Laguerre polynomials in order to compute the Taylor coefficients of the reciprocal of the $R$-transform of the free Jacobi process associated with a single pr…
We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number,in terms of derivatives of powers of an elementary function. The unique explicit expression for the coefficients that appears to be known is that in the book by L. Comtet, which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet's expr…
As further development of earlier works on the $(f,g)$-inversion, the present paper is devoted to the $(f,g)$-difference operator and the representation problem or an expansion formula of analytic functions. A recursive formula and the Leibniz formula for the $(f,g)$-difference operator of the product of two functions are established. The resulting expansion formula not only unifies the $q$-analogue of the Lagrange inversion formula of Gessel and Stanton (thus, a $q$-expansion formula of Liu) for $q$-series but also systematizes the "Ismail's argument". In the meantime, a rigorous analytic pro…
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 polyno…
A {\em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $k\geq 0$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. The present paper studies three kinds of statistics on $k$-generalized Dyck paths: "number of $u$-segments", "number of internal $u$-segments" and "number of $(u,h)$-segments". The Lagrange inversion formula is used to represent the generating function for the number of $k$-generalized Dyck paths accor…
The Δ-convolution of real probability measures, introduced by Bożejko, generalizes both free and boolean convolutions. It is linearized by the Δ-cumulants, and Yoshida gave a combinatorial formula for moments in terms of Δ-cumulants, that implicitly defines the latter. It relies on the definition of an appropriate weight on noncrossing partitions. We give here two different expressions for the Δ-cumulants: the first one is a simple variant of Lagrange inversion formula, and the second one is a combinatorial inversion of Yoshida's formula involving Schröder trees.
We give a short proof, using Lagrange inversion, of a congruence modulo 3 for the number of connected noncrossing graphs on n vertices that was conjectured by Emeric Deutsch and Bruce Sagan. A more complicated proof had been given earlier by S.-P. Eu, S.-C. Liu, and Y.-N. Yeh.
The objective of this paper is, in the main, twofold: Firstly, to develop an algebraic setting for dealing with Bell polynomials and related extensions. Secondly, based on the author's previous work on multivariate Stirling polynomials (2015), to present a number of new results related to different types of inverse relationships, among these (1) the use of multivariable Lah polynomials for characterizing self-orthogonal families of polynomials that can be represented by Bell polynomials, (2) the introduction of `generalized Lagrange inversion polynomials' that invert functions characterized in…
A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.
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 Faà 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…
Two doubly indexed families of homogeneous and isobaric polynomials in several indeterminates are considered: the (partial) exponential Bell polynomials $B_{n,k}$ and a new family $S_{n,k}$ such that $X_1^{-(2n-1)}S_{n,k}$ and $B_{n,k}$ obey an inversion law which generalizes that of the Stirling numbers of the first and second kind. Both polynomial families appear as Lie coefficients in expansions of certain derivatives of higher order. Substituting $D^j(φ)$ (the $j$-th derivative of a fixed function $φ$) in place of the indeterminates $X_j$ shows that both $S_{n,k}$ and $B_{n,k}$ are differe…
This is a journey through integrals of involutions and surprising consequences of the Lagrange inversion theorem. On the way, we meet unexpected logarithmic identities, hypergeometric functions with a linear regime and other mysterious objects. This study was inspired by some results from the fascinating article A.E. Holroyd, T.M. Liggett and D. Romik, Integrals, Partitions, and Cellular Automata, Transactions of the American Mathematical Society, 356-8, 3349-3368, 2004
Let F(x,y) be a function of two variables, and suppose y = f(x) satisfies F(x,y)=0 in some range. Then dy/dx = -Fx/Fy, where Fx and Fy denote the partial derivatives of F with respect to x and y. It is natural to seek a general expression for the higher derivatives d^ny/dx^n, in terms of partial derivatives of F, and such an expression was given in 1974 by L. Comtet and M. Fiolet. Their formula, however, contains some errors. In this note, we give a corrected expression. We give a derivation using Lagrange inversion and also an elementary proof by induction. We further correct a minor error in…
This is a short review on the Faà di Bruno formulas, and some Hopf algebras associated to them. These allow, among several other things, a short proof of the Lie-Scheffers theorem, and relating the Lagrange inversion formulas with antipodes. The link of Faà di Bruno formulas with the theory of set partitions is developed in some detail.
Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. All of our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.
