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.

A General Asymptotic Scheme for the Analysis of Partition Statistics

2014 ◽  
Vol 23 (6) ◽  
pp. 1057-1086 ◽  
Author(s):  
PETER J. GRABNER ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Generating Function ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Higher Moments ◽  
Integer Partitions ◽  
Classical Singularity ◽  
Random Integer ◽  
Partition Statistics ◽  
New Statistics

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

scholarly journals A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

2019 ◽  
Vol 7 ◽  
Author(s):  
DANIEL M. KANE ◽  
ROBERT C. RHOADES
Keyword(s):  
Asymptotic Behavior ◽  
Modular Form ◽  
Generating Function ◽  
Generating Functions ◽  
Asymptotic Properties ◽  
Bootstrap Percolation ◽  
Asymptotic Result ◽  
Integer Partitions ◽  
Mock Modular Form ◽  
Precise Asymptotic

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of$n$without$k$consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without$k$consecutive parts. Andrews showed that when$k=2$, the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For$k>2$, the asymptotic properties of the generating functions have proved more difficult to obtain. Using$q$-series identities and the$k=2$case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case$k=3$was given by Zagier.This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.

scholarly journals Locally Convex Words and Permutations

10.37236/5396 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Christopher Coscia ◽  
Jonathan DeWitt
Keyword(s):  
Generating Functions ◽  
Transfer Matrix Method ◽  
Explicit Solution ◽  
Continued Fraction ◽  
Integer Partitions ◽  
Convexity Condition ◽  
Fixed Size ◽  
Large N ◽  
Locally Convex ◽  
Convexity Parameter

We introduce some new classes of words and permutations characterized by the second difference condition $\pi(i-1) + \pi(i+1) - 2\pi(i) \leq k$, which we call the $k$-convexity condition. We demonstrate that for any sized alphabet and convexity parameter $k$, we may find a generating function which counts $k$-convex words of length $n$. We also determine a formula for the number of 0-convex words on any fixed-size alphabet for sufficiently large $n$ by exhibiting a connection to integer partitions. For permutations, we give an explicit solution in the case $k = 0$ and show that the number of 1-convex and 2-convex permutations of length $n$ are $\Theta(C_1^n)$ and $\Theta(C_2^n)$, respectively, and use the transfer matrix method to give tight bounds on the constants $C_1$ and $C_2$. We also providing generating functions similar to the the continued fraction generating functions studied by Odlyzko and Wilf in the "coins in a fountain" problem.

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 Staircases to Analytic Sum-Sides for Many New Integer Partition Identities of Rogers-Ramanujan Type

10.37236/7847 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Shashank Kanade ◽  
Matthew C. Russell
Keyword(s):  
Lie Algebra ◽  
Integer Partition ◽  
Partition Identities ◽  
Affine Lie Algebra ◽  
Level 2 ◽  
Jagged Partitions

We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the principally specialized characters of certain level $2$ modules for the affine Lie algebra $A_9^{(2)}$. Secondly, we provide analytic sum-sides to some earlier conjectures of the authors. Next, we use these analytic sum-sides to discover a number of further generalizations. Lastly, we apply this technique to the well-known Capparelli identities and present analytic sum-sides which we believe to be new. All of the new conjectures presented in this article are supported by a strong mathematical evidence.  

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.

Durfee squares in compositions

2018 ◽  
Vol 28 (6) ◽  
pp. 359-367 ◽  
Author(s):  
Margaret Archibald ◽  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Asymptotic Analysis ◽  
Generating Functions ◽  
Integer Partitions ◽  
Definition Of ◽  
Durfee Square ◽  
Durfee Squares

Abstract We study compositions (ordered partitions) of n. More particularly, our focus is on the bargraph representation of compositions which include or avoid squares of size s × s. We also extend the definition of a Durfee square (studied in integer partitions) to be the largest square which lies on the base of the bargraph representation of a composition (i.e., is ‘grounded’). Via generating functions and asymptotic analysis, we consider compositions of n whose Durfee squares are of size less than s × s. This is followed by a section on the total and average number of grounded s × s squares. We then count the number of Durfee squares in compositions of n.

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.

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