r-Stirling Numbers Research Articles

Study on Degenerate Stirling Numbers

In this paper, we consider various Stirling numbers of both kinds, including the unsigned degenerate Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, the unsigned degenerate r-Stirling numbers of the first kind and the degenerate r-Stirling numbers of the second kind. The aim of this paper is by using generating functions to further study explicit expressions, some identities and equivalent relations for those Stirling numbers.

On the Total Positivity and Accurate Computations of r-Bell Polynomial Bases

A new class of matrices defined in terms of r-Stirling numbers is introduced. These r-Stirling matrices are totally positive and determine the linear transformation between monomial and r-Bell polynomial bases. An efficient algorithm for the computation to high relative accuracy of the bidiagonal factorization of r-Stirling matrices is provided and used to achieve computations to high relative accuracy for the resolution of relevant algebraic problems with collocation, Wronskian, and Gramian matrices of r-Bell bases. The numerical experimentation confirms the accuracy of the proposed procedure.

R-Lah distribution: Properties, limit theorems and an application to compressed sensing

We introduce and study the r-Lah distribution whose definition involves r-Stirling numbers of both kinds. We compute its expectation and variance, show its log-concavity and prove limit theorems for this distribution. We use these results to prove threshold phenomena for convex cones generated by random walks and to analyze the probability of unique recovery of sparse monotone signals from linear measurements.

Some identities related to degenerate r-Bell and degenerate Fubini polynomials

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler polynomials. This paper is focused on the study of some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind. Especially, we express the power series in terms of the degenerate r-Bell polynomials, of the degenerate r-Stirling numbers of the second kind and of the degenerate Fubini polynomials.

Combinatorial identities involving degenerate harmonic and hyperharmonic numbers

In recent years, some degenerate versions of quite a few special numbers and polynomials are introduced and investigated by means of various methods. The aim of this paper is to study some results on degenerate harmonic numbers, degenerate hyperharmonic numbers, degenerate Fubi polynomials and degenerate r-Fubini polynomials from a general identity which is valid for any two formal power series and involves the degenerate r-Stirling numbers of the second kind.

Asymptotic normality of the Stirling-Whitney-Riordan triangle

Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0 satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 + [(2?b1 + a1)k + a2 + ?(b1 + b2)]Tn?1,k + ?(a1 + ?b1)(k + 1)Tn?1,k+1, where initial conditions Tn,k = 0 unless 0 ? k ? n and T0,0 = 1. Denote by Tn = Pnk =0 Tn,k. In this paper, we show the asymptotic normality of Tn,k and give an asymptotic formula of Tn. As applications, we show the asymptotic normality of many famous combinatorial numbers, such as the Stirling numbers of the second kind, the Whitney numbers of the second kind, the r-Stirling numbers and the r-Whitney numbers of the second kind.

The (l,r)-Stirling numbers: A combinatorial approach

This work deals with a new generalization of r-Stirling numbers using l-tuple of permutations and partitions called (l,r)-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations and symmetric functions. Also, we give a limit representation of the multiple zeta function using (l,r)-Stirling of the first kind.

Some Identities on Degenerate $$r$$-Stirling Numbers via Boson Operators

Broder introduced the r-Stirling numbers of the first kind and of the second kind which enumerate restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively in distinct subsets. Kim-Kim-Lee-Park constructed the degenerate r-Stirling numbers of both kinds as degenerate versions of them. The aim of this paper is to derive some identities and recurrence relations for the degenerate r-Stirling numbers of the first kind and of the second kind via boson operators. In particular, we obtain the normal ordering of a degenerate integral power of the number operator multiplied by an integral power of the creation boson operator in terms of boson operators where the degenerate r-Stirling numbers of the second kind appear as the coefficients.

Riordan arrays and r-Stirling number identities

By using the theory of Riordan arrays, we establish four pairs of general r-Stirling number identities, which reduce to various identities on harmonic numbers, hyperharmonic numbers, the Stirling numbers of the first and second kind, the r-Stirling numbers of the first and second kind, and the r-Lah numbers. We further discuss briefly the connections between the r-Stirling numbers and the Cauchy numbers, the generalized hyperharmonic numbers, and the poly-Bernoulli polynomials. Many known identities are shown to be special cases of our results, and the combinatorial interpretations of several particular identities are also presented as supplements.

Some Identities on λ-analogues of r-Stirling Numbers of the Second Kind

Recently, the λ-analogues of r-Stirling numbers of the first kind were studied by Kim-Kim. The aim of this paper is to introduce the λ-analogues of r-Stirling numbers of the second kind and to investigate some properties, recurrence relations and certain identities on those numbers. We also introduce the λ-analogues of Whitney-type r-Stirling numbers of the second and derive similar results to the case of the λ-analogues of r-Stirling numbers of the second kind. In addition, we consider the λ-analogues of Dowling polynomials and deduce a Dobinski-like formula.

Explicit estimates for Comtet numbers of the first kind

We give explicit upper and lower bounds for a large subset of Comtet numbers sα(n,m) of the first kind, including the r-Stirling numbers of the first kind, among others. In many occasions, such estimates are asymptotically sharp. The form of the bounds varies according to m lying in the central or non-central regions of {1,…,n}. Depending on this fact, we use different probabilistic representations of sα(n,m) in terms of well known random variables.

Note on the Higher-Order Derivatives of the Hyperharmonic Polynomials and the r-Stirling Polynomials of the First Kind

In this paper, we focus on the higher-order derivatives of the hyperharmonic polynomials, which are a generalization of the ordinary harmonic numbers. We determine the hyperharmonic polynomials and their successive derivatives in terms of the r-Stirling polynomials of the first kind and show the relationship between the (exponential) complete Bell polynomials and the r-Stirling numbers of the first kind. Furthermore, we provide a new formula for obtaining the generalized Bernoulli polynomials by exploiting their link with the higher-order derivatives of the hyperharmonic polynomials. In addition, we obtain various identities involving the r-Stirling numbers of the first kind, the Bernoulli numbers and polynomials, the Stirling numbers of the first and second kind, and the harmonic numbers.

English

We give a graph theoretic interpretation of r-Lah numbers, namely, we show that the r-Lah number $${\left\lfloor {\matrix{n \cr k \cr } } \right\rfloor _r}$$ counting the number of r-partitions of an (n + r)-element set into k + r ordered blocks is just equal to the number of matchings consisting of n − k edges in the complete bipartite graph with partite sets of cardinality n and n + 2r − 1 (0 ⩽ k ⩽ n, r ⩾ 1). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for r-Stirling numbers of the second kind.

A three-term recurrence formula for the generalized Bernoulli polynomials

In the present paper, we propose some new explicit formulas of the higher order Daehee polynomials in terms of the generalized r-Stirling and r-Whitney numbers of the second kind. As a consequence, we derive a three-term recurrence formula for the calculation of the generalized Bernoulli polynomials of order k.

A note on degenerate r-Stirling numbers

The aim of this paper is to study the unsigned degenerate r-Stirling numbers of the first kind as degenerate versions of the r-Stirling numbers of the first kind and the degenerate r-Stirling numbers of the second kind as those of the r-Stirling numbers of the second kind. For the degenerate r-Stirling numbers of both kinds, we derive recurrence relations, generating functions, explicit expressions, and some identities involving them.

Some identities on λ-analogues of r-Stirling numbers of the first kind

In this paper, we study ?-analogues of the r-Stirling numbers of the first kind which have close connections with the r-Stirling numbers of the first kind and ?-Stirling numbers of the first kind. Specifically, we give the recurrence relations for these numbers and show their connections with the ?-Stirling numbers of the first kind and higher-order Daehee polynomials.

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

Abstract 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

Mixed restricted Stirling numbers

We investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.

Restricted r-Stirling numbers and their combinatorial applications

We study set partitions with r distinguished elements and block sizes found in an arbitrary index set S. The enumeration of these (S, r)-partitions leads to the introduction of (S, r)-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the r-Stirling numbers. We also introduce the associated (S, r)-Bell and (S, r)-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For S with some extra structure, we show that the inverse of the (S, r)-Stirling matrix encodes the Möbius functions of two families of posets. Through several examples, we demonstrate that for some S the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the (S, r)-Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related (S, r) generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences.

New combinatorial interpretations of [formula omitted]-Whitney and [formula omitted]-Whitney–Lah numbers

T. A. Dowling introduced Whitney numbers of the first and second kinds concerning the so-called Dowling lattices of finite groups. It turned out that they are generalizations of Stirling numbers. Later, I. Mező defined r-Whitney numbers as common generalizations of Whitney numbers and r-Stirling numbers. Additionally, G.-S. Cheon and J.-H. Jung defined r-Whitney–Lah numbers.In our paper, we give new combinatorial interpretations of r-Whitney and r-Whitney–Lah numbers, which correspond better with the combinatorial definitions of Stirling, r-Stirling, Lah and r-Lah numbers. These allow us to explain their properties in a purely combinatorial manner, as well as derive several new identities.