scholarly journals Enumeration of Restricted Permutation Triples

10.37236/403 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Xiaojing Chen ◽  
Wenchang Chu
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Natural Numbers ◽  
Counting Problem ◽  
Restricted Permutation

The counting problem is investigated for the permutation triples of the first $n$ natural numbers with exactly $k$ occurrences of simultaneous "rises". Their recurrence relations and bivariate generating functions are established.

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.

scholarly journals n-Color partitions with weighted differences equal to minus two

1997 ◽  
Vol 20 (4) ◽  
pp. 759-768 ◽  
Author(s):  
A. K. Agarwal ◽  
R. Balasubrananian
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Combinatorial Identities ◽  
Combinatorial Identity ◽  
Divisor Function ◽  
Open Problems

In this paper we study thosen-color partitions of Agarwal and Andrews, 1987, in which each pair of parts has weighted difference equal to−2Results obtained in this paper for these partitions include several combinatorial identities, recurrence relations, generating functions, relationships with the divisor function and computer produced tables. By using these partitions an explicit expression for the sum of the divisors of odd integers is given. It is shown how these partitions arise in the study of conjugate and self-conjugaten-color partitions. A combinatorial identity for self-conjugaten-color partitions is also obtained. We conclude by posing several open problems in the last section.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

scholarly journals Basic Properties of the Polygonal Sequence

2021 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Partial Sums ◽  
Basic Properties ◽  
Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations, fundamental identities, weighted binomial and ordinary sums and the partial sums and generating functions of their powers. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers as is the case in most literature.

scholarly journals Basic Properties of the Polygonal Sequence

2021 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Generating Functions ◽  
Continued Fraction ◽  
Recurrence Relations ◽  
Partial Sums ◽  
Basic Properties ◽  
Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

scholarly journals Note on the Hurwitz–Lerch Zeta Function of Two Variables

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1431
Author(s):  
Junesang Choi ◽  
Recep Şahin ◽  
Oğuz Yağcı ◽  
Dojin Kim
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Zeta Functions ◽  
Integral Representations ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
The One ◽  
Function Of Two Variables

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

scholarly journals Degenerate Stirling Polynomials of the Second Kind and Some Applications

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1046 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Random Variables

Recently, the degenerate λ -Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate λ -Stirling polynomials as well as the r-truncated degenerate λ -Stirling polynomials of the second kind which are derived from generating functions and Newton’s formula. We derive recurrence relations and various expressions for them. Regarding applications, we show that both the degenerate λ -Stirling polynomials of the second and the r-truncated degenerate λ -Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables.

The Enumeration of Maps on the Torus and the Projective Plane

1988 ◽  
Vol 31 (3) ◽  
pp. 257-271 ◽  
Author(s):  
E. A. Bender ◽  
E. R. Canfield ◽  
R. W. Robinson
Keyword(s):  
Projective Plane ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Embedded Graphs ◽  
Asymptotic Expressions ◽  
Explicit Expressions

AbstractThe enumeration of rooted maps (embedded graphs), by number of edges, on the torus and projective plane, is studied. Explicit expressions for the generating functions are obtained. From these are derived asymptotic expressions and recurrence relations. Numerical tables for the numbers with up to 20 edges are presented.

A direct derivation of the Catalan formula

2013 ◽  
Vol 97 (538) ◽  
pp. 53-60 ◽  
Author(s):  
Gerry Leversha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Reflection Principle ◽  
Catalan Numbers ◽  
Equal Size ◽  
Alternative Form ◽  
Direct Derivation ◽  
Image Position

Many readers will be familiar with the sequence of Catalan numbers {Cn: n ≥ 0} and the formulawith its alternative formThese can be proved by using recurrence relations, generating functions or André's reflection principle. A good reference for all of these methods is Martin Griffiths' book [1].However, none of these approaches strikes me as being naturally combinatorial. A formula such as (1) is often derived by making a list of all the ways of doing something, and then subdividing this list into classes of equal size, so that either one class consists entirely of ‘valid’ cases or there is exactly one ‘valid’ case in each list.

scholarly journals General Linear Recurrence Sequences and Their Convolution Formulas

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 132 ◽  
Author(s):  
Paolo Emilio Ricci ◽  
Pierpaolo Natalini
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Second Order ◽  
General Linear ◽  
Linear Recurrence ◽  
Order Linear ◽  
Matrix Polynomials ◽  
Recurrence Sequences ◽  
Straightforward Extension

We extend a technique recently introduced by Chen Zhuoyu and Qi Lan in order to find convolution formulas for second order linear recurrence polynomials generated by 1 1 + a t + b t 2 x . The case of generating functions containing parameters, even in the numerator is considered. Convolution formulas and general recurrence relations are derived. Many illustrative examples and a straightforward extension to the case of matrix polynomials are shown.

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