scholarly journals Rook Theory, Generalized Stirling Numbers and $(p,q)$-Analogues

10.37236/1837 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
J. B. Remmel ◽  
Michelle L. Wachs
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Set Partitions ◽  
Growth Functions ◽  
Rook Theory ◽  
Signed Permutations ◽  
Generalized Stirling Numbers ◽  
Alpha Beta ◽  
Natural Statistics ◽  
Appl Math

In this paper, we define two natural $(p,q)$-analogues of the generalized Stirling numbers of the first and second kind $S^1(\alpha,\beta,r)$ and $S^2(\alpha,\beta,r)$ as introduced by Hsu and Shiue [Adv. in Appl. Math. 20 (1998), 366–384]. We show that in the case where $\beta =0$ and $\alpha$ and $r$ are nonnegative integers both of our $(p,q)$-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our $(p,q)$-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our $(p,q)$-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

scholarly journals The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

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.

On generating functions for the special polynomials

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Special Polynomials ◽  
New Family ◽  
Bernoulli Polynomials And Numbers ◽  
Computation Algorithm ◽  
Generalized Stirling Numbers

The aim of this paper is to investigate and give a new family of Apell type polynomials, which are related to the Euler, Frobenius-Euler and Apostol-Bernoulli polynomials and numbers and also the generalized Stirling numbers of the second kind etc. The results presented in this paper are based upon the theory of the generating functions. By using functional equations of these generating functions, we drive some identities and relations for these numbers and polynomials. Moreover, we give a computation algorithm these numbers.

scholarly journals Q-extensions of some results involving the Luo-Srivastava generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 329-351 ◽  
Author(s):  
Qiu-Ming Luoa
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Duke Math ◽  
Euler Polynomials ◽  
Basic Properties ◽  
In Series ◽  
Appl Math

Carlitz firstly defined the q-Bernoulli and q-Euler polynomials [Duke Math. J., 15 (1948), 987- 1000]. Recently, M. Cenkci and M. Can [Adv. Stud. Contemp. Math., 12 (2006), 213-223], J. Choi, P. J. Anderson and H. M. Srivastava [ Appl. Math. Comput., 199 (2008), 723-737] further defined the q-Apostol-Bernoulli and q-Apostol-Euler polynomials. In this paper, we show the generating functions and basic properties of the q-Apostol-Bernoulli and q-Apostol-Euler polynomials, and obtain some relationships between the q-Apostol-Bernoulli and q-Apostol-Euler polynomials which are the corresponding q-extensions of some known results. Some formulas in series of q-Stirling numbers of the second kind are also considered.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

scholarly journals New parameterized quantum integral inequalities via η-quasiconvexity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Eze R. Nwaeze ◽  
Ana M. Tameru
Keyword(s):  
Integral Inequalities ◽  
Quantum Integral ◽  
Alpha Beta ◽  
Appl Math

Abstract We establish new quantum Hermite–Hadamard and midpoint types inequalities via a parameter $\mu \in [0,1]$ μ ∈ [ 0 , 1 ] for a function F whose $|{}_{\alpha }D_{q}F|^{u}$ | α D q F | u is η-quasiconvex on $[\alpha ,\beta ]$ [ α , β ] with $u\geq 1$ u ≥ 1 . Results obtained in this paper generalize, sharpen, and extend some results in the literature. For example, see (Noor et al. in Appl. Math. Comput. 251:675–679, 2015; Alp et al. in J. King Saud Univ., Sci. 30:193–203, 2018) and (Kunt et al. in Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112:969–992, 2018). By choosing different values of μ, loads of novel estimates can be deduced. We also present some illustrative examples to show how some consequences of our results may be applied to derive more quantum inequalities.

scholarly journals Counting visible levels in bargraphs and set partitions

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6229-6237 ◽  
Author(s):  
Nenad Cakic ◽  
Toufik Mansour ◽  
Rebecca Smith
Keyword(s):  
Generating Functions ◽  
Set Partitions

In this paper, we study the generating functions for the number of visible levels in compositions of n and set partitions of [n].

scholarly journals Unimodality properties of generalized Stirling numbers

2015 ◽  
Vol 45 (9) ◽  
pp. 1583-1586
Author(s):  
Yi WANG ◽  
BaoXuan ZHU ◽  
Lily Li LIU
Keyword(s):  
Stirling Numbers ◽  
Generalized Stirling Numbers

scholarly journals On a Class of Combinatorial Sums Involving Generalized Factorials

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
W.-S. Chou ◽  
L. C. Hsu ◽  
P. J.-S. Shiue
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Sums ◽  
Generalized Stirling Numbers

The object of this paper is to show that generalized Stirling numbers can be effectively used to evaluate a class of combinatorial sums involving generalized factorials.

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