scholarly journals Linked Partition Ideals, Directed Graphs and $q$-Multi-Summations

10.37236/9446 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Shane Chern
Keyword(s):  
Graph Theory ◽  
Generating Function ◽  
Directed Graphs ◽  
Binary Trees ◽  
Computer Assisted ◽  
Integer Partitions ◽  
Factorization Problem ◽  
Difference Systems ◽  
Interesting Connection ◽  
Basic Graph

In this paper, we start by considering generating function identities for linked partition ideals in the setting of basic graph theory. Then our attention is turned to $q$-difference systems, which eventually lead to a factorization problem of a special type of column functional vectors involving $q$-multi-summations. Using a recurrence relation satisfied by certain $q$-multi-summations, we are able to provide non-computer-assisted proofs of some Andrews--Gordon type generating function identities. These proofs also have an interesting connection with binary trees. Further, we give illustrations of constructing a linked partition ideal, or more loosely, a set of integer partitions whose generating function corresponds to a given set of special $q$-multi-summations.

scholarly journals Nonexistence of Almost Moore Digraphs of Diameter Three

10.37236/811 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
J. Conde ◽  
J. Gimbert ◽  
J. Gonzàlez ◽  
J. M. Miret ◽  
R. Moreno
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Algebraic Number ◽  
Directed Graphs ◽  
Algebraic Number Theory ◽  
Cycle Structure ◽  
Spectral Techniques ◽  
Binary Matrices ◽  
Algebraic Multiplicities ◽  
Big Pieces

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound $M(d,k)=1+d+\cdots +d^k$, where $d>1$ and $k>1$ denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when $d=2,3$ or $k=2$. In this paper, we prove that almost Moore digraphs of diameter $k=3$ do not exist for any degree $d$. The enumeration of almost Moore digraphs of degree $d$ and diameter $k=3$ turns out to be equivalent to the search of binary matrices $A$ fulfilling that $AJ=dJ$ and $I+A+A^2+A^3=J+P$, where $J$ denotes the all-one matrix and $P$ is a permutation matrix. We use spectral techniques in order to show that such equation has no $(0,1)$-matrix solutions. More precisely, we obtain the factorization in ${\Bbb Q}[x]$ of the characteristic polynomial of $A$, in terms of the cycle structure of $P$, we compute the trace of $A$ and we derive a contradiction on some algebraic multiplicities of the eigenvalues of $A$. In order to get the factorization of $\det(xI-A)$ we determine when the polynomials $F_n(x)=\Phi_n(1+x+x^2+x^3)$ are irreducible in ${\Bbb Q}[x]$, where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial, since in such case they become 'big pieces' of $\det(xI-A)$. By using concepts and techniques from algebraic number theory, we prove that $F_n(x)$ is always irreducible in ${\Bbb Q}[x]$, unless $n=1,10$. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.

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 Integer Partitions and Binary Trees

2002 ◽  
Vol 28 (3-4) ◽  
pp. 592-601 ◽  
Author(s):  
Frank Schmidt
Keyword(s):  
Binary Trees ◽  
Integer Partitions

THE CUBE-OF-RINGS INTERCONNECTION NETWORK

1998 ◽  
Vol 09 (01) ◽  
pp. 25-37 ◽  
Author(s):  
THOMAS J. CORTINA ◽  
ZHIWEI XU
Keyword(s):  
Graph Theory ◽  
Interconnection Networks ◽  
Fault Tolerant ◽  
Interconnection Network ◽  
Routing Algorithm ◽  
Parallel Computers ◽  
Closed Form Expression ◽  
Form Expression ◽  
Graph Theoretic ◽  
Basic Graph

We present a family of interconnection networks named the Cube-Of-Rings (COR) networks along with their basic graph-theoretic properties. Aspects of group graph theory are used to show the COR networks are symmetric and optimally fault tolerant. We present a closed-form expression of the diameter and optimal one-to-one routing algorithm for any member of the COR family. We also discuss the suitability of the COR networks as the interconnection network of scalable parallel computers.

Directed Graphs Applied to Human Trafficking

2018 ◽  
Vol 14 (03) ◽  
pp. 445-455
Author(s):  
John N. Mordeson ◽  
John H. Mordeson ◽  
Sunil Mathew
Keyword(s):  
Graph Theory ◽  
Mathematical Models ◽  
Human Trafficking ◽  
Directed Graphs ◽  
Over Time

Mathematical models are constructed using directed graphs in order to study the human trafficking flow of male customers and victims. The models are constructed so that the flow can be studied over time. We show that the classes in the models interact with each other in a way one would expect. We study the interaction between the two models. We use results from graph theory to determine the classes which are strengthening, weakening, or neutral members of the directed graphs involving the male customers and victims.

scholarly journals Connections on Valuated Binary Tree and Their Applications in Factoring Odd Integers

2021 ◽  
pp. 134-153
Author(s):  
Xingbo Wang ◽  
Jinfeng Luo ◽  
Ying Tian ◽  
Li Ma
Keyword(s):  
Binary Tree ◽  
Numerical Experiments ◽  
Mathematical Reasoning ◽  
Binary Trees ◽  
Integer Factorization ◽  
Parallel Connection ◽  
Central Lines ◽  
Factorization Problem ◽  
Integer Factorization Problem

This paper makes an investigation on geometric relationships among nodes of the valuated binary trees, including parallelism, connection and penetration. By defining central lines and distance from a node to a line, some intrinsic connections are discovered to connect nodes between different subtrees. It is proved that a node out of a subtree can penetrate into the subtree along a parallel connection. If the connection starts downward from a node that is a multiple of the subtree’s root, then all the nodes on the connection are multiples of the root. Accordingly composite odd integers on such connections can be easily factorized. The paper proves the new results with detail mathematical reasoning and demonstrates several numerical experiments made with Maple software to factorize rapidly a kind of big odd integers that are of the length from 59 to 99 decimal digits. It is once again shown that the valuated binary tree might be a key to unlock the lock of the integer factorization problem.

scholarly journals On Bi-gram Graph Attributes

2021 ◽  
Vol 14 (3) ◽  
pp. 78
Author(s):  
Thomas Konstantinovsky ◽  
Matan Mizrachi
Keyword(s):  
Graph Theory ◽  
Graph Coloring ◽  
Semantic Analysis ◽  
Large Datasets ◽  
Corpus Analysis ◽  
Graph Representation ◽  
New Approach ◽  
Graph Density ◽  
Coloring Graph ◽  
Basic Graph

We propose a new approach to text semantic analysis and general corpus analysis using, as termed in this article, a "bi-gram graph" representation of a corpus. The different attributes derived from graph theory are measured and analyzed as unique insights or against other corpus graphs, attributes such as the graph chromatic number and the graph coloring, graph density and graph K-core. We observe a vast domain of tools and algorithms that can be developed on top of the graph representation; creating such a graph proves to be computationally cheap, and much of the heavy lifting is achieved via basic graph calculations. Furthermore, we showcase the different use-cases for the bi-gram graphs and how scalable it proves to be when dealing with large datasets.

scholarly journals Complexity and polymorphisms for digraph constraint problems under some basic constructions

2016 ◽  
Vol 26 (07) ◽  
pp. 1395-1433 ◽  
Author(s):  
Marcel Jackson ◽  
Tomasz Kowalski ◽  
Todd Niven
Keyword(s):  
Directed Graphs ◽  
Complete Characterization ◽  
Constraint Satisfaction Problems ◽  
Local Consistency ◽  
First Order ◽  
Graph Theoretic ◽  
The Stability ◽  
Basic Graph

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context, we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the “few subpowers algorithm” if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of “strict width” and solvability by few subpowers are unstable under first-order reductions. The analysis also yields a complete characterization of the main polymorphism properties for digraphs whose symmetric closure is a complete graph.

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