scholarly journals On Certain Combinatorial Expansions of the Legendre-Stirling Numbers

10.37236/8060 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Shi-Mei Ma ◽  
Jun Ma ◽  
Yeong-Nan Yeh
Keyword(s):  
Spectral Theory ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Set Partitions ◽  
Combinatorial Code

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. As a continuation of the work of Andrews and Littlejohn (Proc. Amer. Math. Soc., 137 (2009), 2581-2590), we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain expansions of the Legendre-Stirling numbers of both kinds in terms of binomial coefficients.

scholarly journals Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 112 ◽  
Author(s):  
Irem Kucukoglu ◽  
Burcin Simsek ◽  
Yilmaz Simsek
Keyword(s):  
Probability Distribution ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Distribution Functions ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Charlier Polynomials ◽  
Combinatorial Sums

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

scholarly journals Set partitions with successions and separations

2005 ◽  
Vol 2005 (3) ◽  
pp. 451-463 ◽  
Author(s):  
Augustine O. Munagi
Keyword(s):  
Stirling Numbers ◽  
Set Partitions ◽  
Consecutive Integers

Partitions of the set{1,2,…,n}are classified as having successions if a block contains consecutive integers, and separated otherwise. This paper constructs enumeration formulas for such set partitions and some variations using Stirling numbers of the second kind.

scholarly journals Counting Pattern-free Set Partitions I: A Generalization of Stirling Numbers of the Second Kind

2000 ◽  
Vol 21 (3) ◽  
pp. 367-378 ◽  
Author(s):  
Martin Klazar
Keyword(s):  
Stirling Numbers ◽  
Set Partitions ◽  
Free Set

scholarly journals Unimodality Problems of Multinomial Coefficients and Symmetric Functions

10.37236/560 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Xun-Tuan Su ◽  
Yi Wang ◽  
Yeong-Nan Yeh
Keyword(s):  
Symmetric Functions ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Early Results ◽  
Multinomial Coefficients

In this note we consider unimodality problems of sequences of multinomial coefficients and symmetric functions. The results presented here generalize our early results for binomial coefficients. We also give an answer to a question of Sagan about strong $q$-log-concavity of certain sequences of symmetric functions, which can unify many known results for $q$-binomial coefficients and $q$-Stirling numbers of two kinds.

scholarly journals Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions

10.37236/2131 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Pietro Mongelli
Keyword(s):  
Spectral Theory ◽  
Symmetric Functions ◽  
Stirling Numbers ◽  
Elementary Symmetric Functions ◽  
Combinatorial Interpretations

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial  interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.

scholarly journals Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1436
Author(s):  
Talha Usman ◽  
Mohd Saif ◽  
Junesang Choi
Keyword(s):  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Research Subjects

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.

scholarly journals Enumerating set partitions by the number of positions between adjacent occurrences of a letter

2010 ◽  
Vol 4 (2) ◽  
pp. 284-308 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Asymptotic Estimates ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Minimal Elements

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

A Unified Interpretation of the Binomial Coefficients, the Stirling Numbers, and the Gaussian Coefficients

2000 ◽  
Vol 107 (10) ◽  
pp. 901 ◽  
Author(s):  
John Konvalina
Keyword(s):  
Stirling Numbers ◽  
Binomial Coefficients

scholarly journals Generalized Stirling Numbers and Hyper-Sums of Powers of Binomials Coefficients

10.37236/3787 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Claudio de J. Pita-Ruiz V.
Keyword(s):  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Partial Sums ◽  
Linear Combinations ◽  
Normal Ordering ◽  
Sums Of Powers ◽  
Generalized Stirling Numbers ◽  
Natural Way

We work with a generalization of Stirling numbers of the second kind related to the boson normal ordering problem (P. Blasiak et al.). We show that these numbers appear as part of the coefficients of expressions in which certain sequences of products of binomials, together with their partial sums, are written as linear combinations of some other binomials. We show that the number arrays formed by these coefficients can be seen as natural generalizations of Pascal and Lucas triangles, since many of the known properties on rows, columns, falling diagonals and rising diagonals in Pascal and Lucas triangles, are also valid (some natural generalizations of them) in the arrays considered in this work. We also show that certain closed formulas for hyper-sums of powers of binomial coefficients appear in a natural way in these arrays.

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