scholarly journals POLYNOMIAL IDENTITIES OF THE ROGERS-RAMANUJAN TYPE

1995 ◽  
Vol 10 (16) ◽  
pp. 2291-2315 ◽  
Author(s):  
OMAR FODA ◽  
YAS-HIRO QUANO
Keyword(s):  
Generating Functions ◽  
Polynomial Identities ◽  
Restricted Partitions ◽  
One To One

Presented are polynomial identities which imply generalizations of Euler and Rogers-Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by establishing a graphical one-to-one correspondence between those two kinds of restricted partitions.

scholarly journals An Infinite Family of Graphs with the Same Ihara Zeta Function

10.37236/354 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Christopher Storm
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Infinite Family ◽  
Ihara Zeta Function ◽  
One To One

In 2009, Cooper presented an infinite family of pairs of graphs which were conjectured to have the same Ihara zeta function. We give a proof of this result by using generating functions to establish a one-to-one correspondence between cycles of the same length without backtracking or tails in the graphs Cooper proposed. Our method is flexible enough that we are able to generalize Cooper's graphs, and we demonstrate additional families of pairs of graphs which share the same zeta function.

scholarly journals Priority Queues and Multisets

10.37236/1218 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
M. D. Atkinson ◽  
S. A. Linton ◽  
L. A. Walker
Keyword(s):  
Data Structure ◽  
Generating Functions ◽  
Priority Queue ◽  
Priority Queues ◽  
A Priority ◽  
One To One ◽  
Output Component ◽  
Fixed Input ◽  
Symmetry Result

A priority queue, a container data structure equipped with the operations insert and delete-minimum, can re-order its input in various ways, depending both on the input and on the sequence of operations used. If a given input $\sigma$ can produce a particular output $\tau$ then $(\sigma,\tau)$ is said to be an allowable pair. It is shown that allowable pairs on a fixed multiset are in one-to-one correspondence with certain k-way trees and, consequently, the allowable pairs can be enumerated. Algorithms are presented for determining the number of allowable pairs with a fixed input component, or with a fixed output component. Finally, generating functions are used to study the maximum number of output components with a fixed input component, and a symmetry result is derived.

scholarly journals Some partition and analytical identities arising from the Alladi, Andrews, Gordon bijection

2020 ◽  
Author(s):  
S. Capparelli ◽  
A. Del Fra ◽  
P. Mercuri ◽  
A. Vietri
Keyword(s):  
Generating Functions ◽  
General Setting ◽  
Integer Partitions ◽  
Partition Identities ◽  
General Kind ◽  
One To One ◽  
Jagged Partitions

Abstract In the work of Alladi et al. (J Algebra 174:636–658, 1995) the authors provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a one-to-one correspondence by the method described in the 1995 paper. As a further result, although of a different nature, we obtain an analytical identity of Rogers–Ramanujan type, involving generating functions, for a class of partition identities already found in that paper and that generalize the first Capparelli identity and include it as a particular case. To achieve this, we apply the same strategy as Kanade and Russell did in a recent paper. This method relies on the use of jagged partitions that can be seen as a more general kind of integer partitions.

Partial Differentiation

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Nineteenth Century ◽  
Generating Functions ◽  
The Other ◽  
Legendre Transform ◽  
Mathematical Tool ◽  
Partial Differentiation ◽  
Thermodynamic Potentials ◽  
One To One

This short chapter discusses the Legendre transform, which is used in mechanics to convert between the Lagrangian and the Hamiltonian formulations. The Legendre transform is a mathematical tool that can be used to convert the variables of a function through the methods of partial differentiation in a one-to-one fashion. Developed by Adrien-Marie Legendre in the nineteenth century, it is also central to converting between action principles, generating functions and thermodynamic potentials. By using the Legendre transform, two variables can be expressed in four different ways, via the idea of conjugate pairs; it just depends on what differential quantity is subtracted. Variables that are not considered in the transformation are called passive variables, whiles the important ones are the active variables. The information in this chapter provides the background for many of the other chapters in this book.

scholarly journals Distances in random Apollonian network structures

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Olivier Bodini ◽  
Alexis Darrasse ◽  
Michèle Soria
Keyword(s):  
Limit Distribution ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Network Structures ◽  
Average Value ◽  
International Audience ◽  
One To One ◽  
Rayleigh Limit

International audience In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we prove a Rayleigh limit distribution for distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order $n$ is asymptotically $\sqrt{n}$. Nous étudions dans ce papier la distribution des distances dans les structures des réseaux apolloniens aléatoires (RANS), une famille de graphes en bijection avec les arbres ternaires planaires. En s'appuyant sur l'utilisation de séries génératrices multivariées pour décrire toute l'information sur les distances, ainsi que sur l'analyse de singularités pour évaluer les coefficients de ces séries, nous prouvons une distribution limite de Rayleigh pour les distances vers un sommet externe du RANS et montrons que la distance moyenne entre deux sommets quelconques d'un RANS d'ordre $n$ est asymptotiquement $\sqrt{n}$.

scholarly journals New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1484
Author(s):  
Yilmaz Simsek
Keyword(s):  
Functional Equation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Polynomial Identities ◽  
Euler Numbers ◽  
Special Polynomial ◽  
Combinatorial Sums ◽  
Changhee Numbers ◽  
Euler’S Identity

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper.

scholarly journals The application of substitutional analysis to invariants

1935 ◽  
Vol 234 (733) ◽  
pp. 79-114 ◽  
Keyword(s):  
Characteristic Function ◽  
Finite Order ◽  
Generating Function ◽  
Generating Functions ◽  
Polynomial Function ◽  
General Theorem ◽  
Simple Extension ◽  
New Form ◽  
One To One ◽  
New System

In a previous paper it was proved that the generating function for any class of ternary concomitants might be obtained from the corresponding generating function for gradients (coefficient products) by multiplication by (1 — x ) (1 — y ) x — y ). A generating function for ternary gradients was given in Theorem III of that paper, but it is of such a character that it is useless for purposes of calculation.In this paper a new system of generating functions is obtained applicable to perpetuants or to forms of finite order, and also to binary, ternary, or any forms. In Section I a class of polynomial function f a ( z ) is discussed which appeared in the paper just quoted in connection with the generating function for binary gradients of particular substitutional form in the perpetuant case. In Section II a one to one correspondence between binary perpetuants of particular substitutional form and the terms of the corresponding generating function is obtained by means of the tableau notation ; and this is used to give a very simple extension of GRACE's Theorem on irreducible perpetuants to the case of perpetuants of particular substitutional form. Section III deals with the properties of the functions for forms of finite order which correspond to the functions f a ( z ) for perpetuants. The generating functions for the different substitutional classes must by addition give the generating function for types. This in some cases has been obtained independently. Thus there arise certain algebraic identities. In Section IV a general theorem is established covering all these identities. It is obtained by means of the Characteristic Function of SCHUR. The same method is then used to express the binary generating functions in a new form.

scholarly journals Forests and pattern-avoiding permutations modulo pure descents

2018 ◽  
Vol 27 (1) ◽  
pp. 18-31
Author(s):  
Jean-Luc Baril ◽  
Sergey Kirgizov ◽  
Armen Petrossian
Keyword(s):  
Equivalence Relation ◽  
Generating Functions ◽  
Equivalence Classes ◽  
Binary Trees ◽  
Square Lattice ◽  
Single Source ◽  
One To One ◽  
Pattern Avoiding Permutations

Abstract We investigate an equivalence relation on permutations based on the pure descent statistic. Generating functions are given for the number of equivalence classes for the set of all permutations, and the sets of permutations avoiding exactly one pattern of length three. As a byproduct, we exhibit a permutation set in one-to-one correspondence with forests of ordered binary trees, which provides a new combinatorial class enumerated by the single-source directed animals on the square lattice. Furthermore, bivariate generating functions for these sets are given according to various statistics.

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