scholarly journals On Subtree Number Index of Generalized Book Graphs, Fan Graphs, and Wheel Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Daoqiang Sun ◽  
Zhengying Zhao ◽  
Xiaoxiao Li ◽  
Jiayi Cao ◽  
Yu Yang
Keyword(s):  
Structural Analysis ◽  
Structural Properties ◽  
Generating Function ◽  
Generating Functions ◽  
Asymptotic Properties ◽  
Wheel Graphs

With generating function and structural analysis, this paper presents the subtree generating functions and the subtree number index of generalized book graphs, generalized fan graphs, and generalized wheel graphs, respectively. As an application, this paper also briefly studies the subtree number index and the asymptotic properties of the subtree densities in regular book graphs, regular fan graphs, and regular wheel graphs. The results provide the basis for studying novel structural properties of the graphs generated by generalized book graphs, fan graphs, and wheel graphs from the perspective of the subtree number index.

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.

A general construction of mutually orthogonal latin squares of prime order

2018 ◽  
Vol 7 (1-2) ◽  
pp. 77-93
Author(s):  
J. A. Saka ◽  
O. O. Oyadare
Keyword(s):  
Structural Properties ◽  
Generating Function ◽  
Generating Functions ◽  
Prime Order ◽  
Latin Squares ◽  
General Construction ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set ◽  
General Method

This paper presents a general method of constructing a complete set of Mutually Orthogonal Latin Squares (MOLS) of the order of any prime, via the use of generating functions dened on the nite eld of this order. Apart from using the generating function to get a complete set of Mutually Orthogonal Latin Squares, the studies of the generating functions opens up the possibility of getting at the deep structural properties of MOLS. Copious examples were given for detailed illustrations.

scholarly journals On BC-Subtrees in Multi-Fan and Multi-Wheel Graphs

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 36
Author(s):  
Yu Yang ◽  
Long Li ◽  
Wenhu Wang ◽  
Hua Wang
Keyword(s):  
Structural Properties ◽  
Generating Functions ◽  
Topological Index ◽  
Extremal Problems ◽  
Distance Constraint ◽  
Wheel Graphs ◽  
Recursive Formulas ◽  
Cyclic Graphs ◽  
New Perspective

The BC-subtree (a subtree in which any two leaves are at even distance apart) number index is the total number of non-empty BC-subtrees of a graph, and is defined as a counting-based topological index that incorporates the leaf distance constraint. In this paper, we provide recursive formulas for computing the BC-subtree generating functions of multi-fan and multi-wheel graphs. As an application, we obtain the BC-subtree numbers of multi-fan graphs, r multi-fan graphs, multi-wheel (wheel) graphs, and discuss the change of the BC-subtree numbers between different multi-fan or multi-wheel graphs. We also consider the behavior of the BC-subtree number in these structures through the study of extremal problems and BC-subtree density. Our study offers a new perspective on understanding new structural properties of cyclic graphs.

Semantic And Structural Properties Of The Banking And Financial Terminology Of The Uzbek Language

2020 ◽  
Vol 2 (08) ◽  
pp. 610-617
Author(s):  
Ekaterina Turakulovna Shirinova
Keyword(s):  
Structural Analysis ◽  
Structural Properties ◽  
Human Factor

This article discusses information on the study of terminology in Uzbek and world linguistics. Thematic grouping of banking and financial terms, which play an important role in Uzbek language vocabulary, is considered. The author gives the criteria for the distribution of terms into thematic groups, their peculiar properties examples to substantiate the hypothesis. The paradigmatic relations between the terms of this sphere are indicated. A structural analysis of the banking and financial terms of the Uzbek language is carried out.  On the basis of the anthropocentric approach, the role of the human factor in the banking and financial terminology of the Uzbek language is studied. Cognitive metaphors that exist in the terminology are considered.

scholarly journals A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1161
Author(s):  
Hari Mohan Srivastava ◽  
Sama Arjika
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Additional Parameter ◽  
Physical Sciences ◽  
General Family ◽  
Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

scholarly journals Generating Functions for Certain Weighted Cranks

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Shreejit Bandyopadhyay ◽  
Ae Yee
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Colored Partitions ◽  
Unimodal Sequences

Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this paper, we first consider the crank generating function weighted by the number of ones and obtain analytic formulas for this weighted crank function under conditions of the crank being less than or equal to some specific integer. We connect these cumulative and point crank functions to the generating functions of partitions with certain sizes of Durfee rectangles. We then consider a generalization of the crank for $k$-colored partitions, which was first introduced by Fu and Tang, and investigate the corresponding generating function for this crank weighted by the number of parts in the first subpartition of a $k$-colored partition. We show that the cumulative generating functions are the same as the generating functions for certain unimodal sequences.

scholarly journals Labels distance in bucket recursive trees with variable capacities of buckets

2021 ◽  
Vol 13 (2) ◽  
pp. 413-426
Author(s):  
S. Naderi ◽  
R. Kazemi ◽  
M. H. Behzadi
Keyword(s):  
Partial Differential Equations ◽  
Probability Distribution ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Function Approach ◽  
Closed Formula ◽  
Tree Model ◽  
Partial Differential ◽  
Recursive Trees

Abstract The bucket recursive tree is a natural multivariate structure. In this paper, we apply a trivariate generating function approach for studying of the depth and distance quantities in this tree model with variable bucket capacities and give a closed formula for the probability distribution, the expectation and the variance. We show as j → ∞, lim-iting distributions are Gaussian. The results are obtained by presenting partial differential equations for moment generating functions and solving them.

scholarly journals Catalan words avoiding pairs of length three patterns

2021 ◽  
Vol vol. 22 no. 2, Permutation... (Special issues) ◽  
Author(s):  
Jean-Luc Baril ◽  
Carine Khalil ◽  
Vincent Vajnovszki
Keyword(s):  
Structural Properties ◽  
Generating Functions ◽  
Recursive Decomposition ◽  
Integer Sequence ◽  
Combinatorial Interpretations

Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.

COUNTING NUMERICAL SEMIGROUPS WITH SHORT GENERATING FUNCTIONS

2011 ◽  
Vol 21 (07) ◽  
pp. 1217-1235 ◽  
Author(s):  
VÍCTOR BLANCO ◽  
PEDRO A. GARCÍA-SÁNCHEZ ◽  
JUSTO PUERTO
Keyword(s):  
Generating Function ◽  
Polynomial Time ◽  
Generating Functions ◽  
Time Complexity ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Polynomial Time Complexity ◽  
Theoretical Results

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.

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.

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