scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

A generalized class of restricted Stirling and Lah numbers

2018 ◽  
Vol 68 (4) ◽  
pp. 727-740 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Properties ◽  
Exponential Generating Function ◽  
Cauchy Numbers ◽  
Restricted Stirling Numbers

Abstract In this paper, we consider a polynomial generalization, denoted by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

scholarly journals A Newton Interpolation Approach to Generalized Stirling Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
Explicit Expression ◽  
Recurrence Relation ◽  
Generating Function ◽  
Stirling Numbers ◽  
Divided Differences ◽  
Newton Interpolation ◽  
Basic Properties ◽  
Generalized Stirling Numbers

We employ the generalized factorials to define a Stirling-type pair{s(n,k;α,β,r),S(n,k;α,β,r)}which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.

scholarly journals Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6879-6891
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bell Numbers ◽  
New Family ◽  
Derivative Formulas ◽  
Combinatorial Sums ◽  
Computation Algorithm ◽  
Fractional Derivative Operators ◽  
And Inversion

The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

2013 ◽  
Vol 11 (5) ◽  
Author(s):  
István Mező
Keyword(s):  
Hypergeometric Function ◽  
Generating Function ◽  
Arithmetic Progressions ◽  
Harmonic Numbers ◽  
Bell Numbers ◽  
Exponential Generating Function ◽  
Hyperharmonic Numbers

AbstractThere is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear.

scholarly journals A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Miguel Méndez ◽  
Adolfo Rodríguez
Keyword(s):  
Stirling Numbers ◽  
Direct Application ◽  
Bell Numbers ◽  
Bijective Correspondence ◽  
Rook Placements ◽  
Combinatorial Model ◽  
International Audience ◽  
Generalized Stirling Numbers ◽  
Dual Graded Graphs ◽  
Combinatorial Methods

International audience We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.

scholarly journals On (q, r, w)-Stirling Numbers of the Second Kind

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Exponential Generating Function

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.

scholarly journals Elliptic Rook and File Numbers

10.37236/6121 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Michael J. Schlosser ◽  
Meesue Yoo
Keyword(s):  
Stirling Numbers ◽  
Classical Case ◽  
Factorization Theorem ◽  
Rook Numbers ◽  
Generalized Stirling Numbers ◽  
Independent Parameters ◽  
Ferrers Boards

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's $q$-rook numbers by two additional independent parameters $a$ and $b$, and a nome $p$. The elliptic rook numbers are shown to satisfy an elliptic extension of a  factorization theorem which in the classical case was established by Goldman, Joichi and White and extended to the $q$-case by Garsia and Remmel. We obtain similar results for elliptic analogues of Garsia and Remmel's $q$-file numbers for skyline boards. We also provide an elliptic extension of the $j$-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and $r$-restricted versions thereof.

scholarly journals The Hankel Transform ofq-Noncentral Bell Numbers

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Cristina B. Corcino ◽  
Roberto B. Corcino ◽  
Jay M. Ontolan ◽  
Charrymae M. Perez-Fernandez ◽  
Ednelyn R. Cantallopez
Keyword(s):  
Stirling Numbers ◽  
Hankel Transform ◽  
Convolution Type ◽  
Combinatorial Interpretation ◽  
Bell Numbers

We define two forms ofq-analogue of noncentral Stirling numbers of the second kind and obtain some properties parallel to those of noncentral Stirling numbers. Certain combinatorial interpretation is given for the second form of theq-analogue in the context of 0-1 tableaux which, consequently, yields certain additive identity and some convolution-type formulas. Finally, aq-analogue of noncentral Bell numbers is defined and its Hankel transform is established.

scholarly journals A $p,q$-analogue of a Formula of Frobenius

10.37236/1702 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Karen S. Briggs ◽  
Jeffrey B. Remmel
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Eulerian Polynomials ◽  
Major Index ◽  
Rook Placements ◽  
Rook Numbers ◽  
Nonnegative Coefficients ◽  
Ferrers Boards

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations to give a combinatorial interpretation to the $q$-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs defined generalized $p,q$-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definition to give a $p,q$-analogue of rook numbers for arbitrary Ferrers boards. In this paper, we use Remmel and Wach's definition and an extension of Garsia and Remmel's proof to give a combinatorial interpretation to a $p,q$-analogue of a formula of Frobenius relating the $p,q$-Stirling numbers of the second kind to the trivariate distribution of the descent number, major index, and comajor index over $S_n$. We further define a $p,q$-analogue of the hit numbers, and show analytically that for Ferrers boards, the $p,q$-hit numbers are polynomials in $(p,q)$ with nonnegative coefficients.

scholarly journals A Colorful Involution for the Generating Function for Signed Stirling Numbers of the First Kind

10.37236/451 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Paul Levande
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Interpretation

We show how the generating function for signed Stirling numbers of the first kind can be proved using the involution principle and a natural combinatorial interpretation based on cycle-colored permuations.

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