Jean-Christophe Novelli, Jean-Yves Thibon
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.
Lagrange Inversion Formula by Induction
本文针对拉格朗日反演公式，提出了一种简洁的归纳法证明方式，为该公式提供了更易理解的推导路径。
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.
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 polyn…
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 give presentation of composition inverse of formal power serie in a logarithmic form.
Some applications of the Lagrange inversion formula for the $k$-Fibonacci numbers
本文利用Lagrange反演公式及Hermite的推论，推导出$k$-Fibonacci数的求和公式及其以广义二项式系数表示的渐近等价式。
Non-external Proofs of Lagrange Inversion Formula
本文给出形式幂级数拉格朗日反演公式的两个简单证明，仅使用形式幂级数分析的基本工具，不依赖外部概念。
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 provide yet another proof of the classical Lagrange-Good multivariable inversion formula using techniques of quantum field theory.
Arnol'd's limit and the Lagrange inversion
本研究利用拉格朗日反演证明了Arnol'd提出的三角函数与反三角函数复合表达式的x→0极限为1，并进一步得到形式幂级数层面的更一般性结论。
Application of Lagrange inversion to wall-crossing for Quot schemes on surfaces
本文利用拉格朗日反演技巧证明了分别由几何虚拟局部化与顶点代数得到的两个幂级数相等，从而使 Quot 概形计数不变量工作独立于先前文献，并给出新的组合恒等式。
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…
Algebraic topology of the Lagrange inversion
本文通过复射影空间的Chern数给出了Lagrange倒置公式的拓扑解释，并引入由Catalan数定义的新形式群，探讨了代数簇的Chern数与欧拉示性之间的关系。
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…
Let $K$ be any unital commutative $\bQ$-algebra and $z=(z_1, z_2, ..., z_n)$ commutative or noncommutative variables. Let $t$ be a formal central parameter and $\kttzz$ the formal power series algebra of $z$ over $K[[t]]$. In \cite{GTS-II}, for each automorphism $F_t(z)=z-H_t(z)$ of $\kttzz$ with $H_{t=0}(z)=0$ and $o(H(z))\geq 1$, a \cNcs (noncommutative symmetric) system (\cite{GTS-I}) $\Oft$ has been constructed. Consequently, we get a Hopf algebra homomorphism $\cSft: \cNsf \to \cDzz$ from the Hopf algebra $\cNsf$ (\cite{G-T}) of NCSF's (noncommutative symmetric functions). In this paper, …
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…
Using the Lagrange inversion formula, $t$-ary trees are enumerated with respect to edge type (left, middle, right for ternary trees).
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…
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.
We investigate the combinatorial properties of the functional equation $ϕ[h(z)]=h(qz)$ for the conjugation of a formal diffeomorphism $ϕ$ of $\mathbb{C}$ to its linear part $z\mapsto qz$. This is done by interpreting the functional equation in terms of symmetric functions, and then lifting it to noncommutative symmetric functions. We describe explicitly the expansion of the solution in terms of plane trees and prove that its expression on the ribbon basis has coefficients in ${\mathbb N}[q]$ after clearing the denominators $(q)_n$. We show that the conjugacy equation can be lifted to a quadrat…
This paper is the first of a sequence papers ([Z4]--[Z7]) on the {\it ${\mathcal N}$CS $(\text{noncommutative symmetric})$ systems} over differential operator algebras in commutative or noncommutative variables ([Z4]); the ${\mathcal N}$CS systems over the Grossman-Larson Hopf algebras ([GL],[F]) of labeled rooted trees ([Z6]); as well as their connections and applications to the inversion problem ([BCW],[E4]) and specializations of NCSFs ([Z5],[Z7]). In this paper, inspired by the seminal work [GKLLRT] on NCSFs (noncommutative symmetric functions), we first formulate the notion {\it ${\mathca…
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
The 0-Hecke algebra $H_n(0)$ is a deformation of the group algebra of the symmetric group $\SS_n$. We show that its coinvariant algebra naturally carries the regular representation of $H_n(0)$, giving an analogue of the well-known result for $\SS_n$ by Chevalley-Shephard-Todd. By investigating the action of $H_n(0)$ on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of $H_n(0)$ on the cohomology rings of the Springer fibers, and similarly interpret…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。