Normally ordered forms of powers of differential operators and their combinatorics

Briand, Emmanuel; Lopes, Samuel A.; Celis, Mercedes Helena Rosas

doi:10.1016/j.jpaa.2020.106312

Cited by 5 publications

(10 citation statements)

References 15 publications

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“…The numbers appearing in A n,k as coefficients can be found in [33, A139605]. We refer the reader to [4,27,28] for various results and examples on the expansions of (cD) n . In 1973, Comtet obtained the following result.…”

Section: Preliminarymentioning

confidence: 99%

Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux

Han¹,

Ma²

2020

Preprint

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In this paper we first present summation formulas for k-order Eulerian polynomials and 1/k-Eulerian polynomials. We then present combinatorial expansions of (c(x)D) n in terms of inversion sequences as well as k-Young tableaux, where c(x) is a differentiable function in the indeterminate x and D is the derivative with respect to x. We define the g-indexes of k-Young tableaux and Young tableaux, which have important applications in combinatorics.By establishing some relations between k-Young tableaux and standard Young tableaux, we express Eulerian polynomials, second-order Eulerian polynomials, André polynomials and the generating polynomials of gamma coefficients of Eulerian polynomials in terms of standard Young tableaux, which imply a deep connection among these polynomials.

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Section: Preliminarymentioning

confidence: 99%

Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux

Han¹,

Ma²

2020

Preprint

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“…Recently, Briand-Lopes-Rosas [11] studied the normally ordered form of the operator cD d . In [11,Corollary 3.11], they presented a generalization of Comtet's explicit formula for F n,k .…”

Section: 1mentioning

confidence: 99%

“…Recently, Briand-Lopes-Rosas [11] studied the normally ordered form of the operator cD d . In [11,Corollary 3.11], they presented a generalization of Comtet's explicit formula for F n,k . Moreover, they gave a survey on the combinatorial and arithmetic properties of the coefficients F n,k , which can be summarized as follows.…”

Section: 1mentioning

confidence: 99%

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Several variants of the Dumont differential system and permutation statistics

Mansour

Wang

et al. 2018

Sci. China Math.

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The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (Math Comp, 1979, 33: 1293-1297 and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme for the cycle structure of permutations. We then introduce two types of Jacobi-pairs of differential equations. We present a general method to derive the solutions of these differential equations. As applications, we present some characterizations for several permutation statistics.

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“…(ii) Computations stemming from work in representation theory initiated in [16] led us generalize a lot of the combinatorics associated to the Weyl algebra in [26].…”

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confidence: 99%