Some Combinatorial Arrays Related to the Lotka-Volterra System

Callan, David; Ma, Shi-Mei; Mansour, Toufik

doi:10.37236/4413

Cited by 2 publications

(2 citation statements)

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“…where f(x) is a formal power series. For more applications of the context-free grammar method see for example [5,7,13,17]. The grammar formal derivative D can be equivalently written as a derivation in the algebra of formal power series over a ring R containing Q,…”

Section: Introductionmentioning

confidence: 99%

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A new approach to the r-Whitney numbers by using combinatorial differential calculus

Méndez

Ramírez

2019

Acta Universitatis Sapientiae, Mathematica

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In the present article we introduce two new combinatorial interpretations of the r-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar G := {y → yxm, x → x}. By specializing m = 1 we obtain also a new combinatorial interpretation of the r-Stirling numbers of the second kind. Again, by specializing to the case r = 0 we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard’s polynomials. Moreover, we recover several known identities involving the r-Dowling polynomials and the r-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The r-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the r-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce [m]-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities

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Section: Introductionmentioning

confidence: 99%

“…where f (x) is a formal power series. For more applications of the context-free grammar method see for example [5,7,11,15,16,17].…”

Section: Introductionmentioning

confidence: 99%