The concrete tetrahedron: symbolic sums, recurrence equations, generating functions, asymptotic estimates

2012 ◽  
Vol 49 (11) ◽  
pp. 49-6334-49-6334
Keyword(s):  
Generating Functions ◽  
Asymptotic Estimates ◽  
Recurrence Equations

scholarly journals Enumerating set partitions by the number of positions between adjacent occurrences of a letter

2010 ◽  
Vol 4 (2) ◽  
pp. 284-308 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Asymptotic Estimates ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Minimal Elements

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

scholarly journals Lindelöf Representations and (Non-)Holonomic Sequences

10.37236/275 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Philippe Flajolet ◽  
Stefan Gerhold ◽  
Bruno Salvy
Keyword(s):  
Finite Difference ◽  
Generating Functions ◽  
Complex Analysis ◽  
Integral Representations ◽  
Asymptotic Estimates ◽  
Linear Recurrences ◽  
Polynomial Coefficients ◽  
Difference Sequences ◽  
Analytic Expressions ◽  
Linear Differential

Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindelöf, which belong to an attractive but somewhat neglected chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotic estimates of certain finite difference sequences come out as a byproduct of the Lindelöf approach.

scholarly journals The Number of Graphs Not Containing $K_{3,3}$ as a Minor

10.37236/838 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Stefanie Gerke ◽  
Omer Giménez ◽  
Marc Noy ◽  
Andreas Weißl
Keyword(s):  
Generating Functions ◽  
Singularity Analysis ◽  
Precise Estimate ◽  
Asymptotic Estimates ◽  
Limit Laws ◽  
A Minor ◽  
Minor Free Graphs ◽  
Precise Asymptotic ◽  
Labelled Graphs

We derive precise asymptotic estimates for the number of labelled graphs not containing $K_{3,3}$ as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random $K_{3,3}$-minor-free graphs, like the number of edges. To establish these results, we translate a decomposition for the corresponding graphs into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph $K_{3,3}$ plus an edge as a minor.

scholarly journals Degree distribution in random planar graphs

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Omer Gimenez ◽  
Marc Noy
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Degree Distribution ◽  
Planar Graphs ◽  
Probability Generating Function ◽  
Asymptotic Estimates ◽  
Asymptotic Enumeration ◽  
Root Vertex ◽  
International Audience ◽  
Transfer Theorems

International audience We prove that for each $k \geq 0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimènez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function $p(w)=\sum_k d_k w^k$. From the explicit expression for $p(w)$ we can compute the $d_k$ to any degree of accuracy, and derive asymptotic estimates for large values of $k$.

On Average Behaviour of Regular Expressions in Strong Star Normal Form

2019 ◽  
Vol 30 (06n07) ◽  
pp. 899-920 ◽  
Author(s):  
Sabine Broda ◽  
António Machiavelo ◽  
Nelma Moreira ◽  
Rogério Reis
Keyword(s):  
Normal Form ◽  
Generating Functions ◽  
Finite Automata ◽  
Regular Expressions ◽  
Large Set ◽  
Asymptotic Estimates ◽  
Analytic Combinatorics ◽  
Average Complexity ◽  
Average Behaviour ◽  
Puiseux Expansions

For regular expressions in (strong) star normal form a large set of efficient algorithms is known, from conversions into finite automata to characterisations of unambiguity. In this paper we study the average complexity of this class of expressions using analytic combinatorics. As it is not always feasible to obtain explicit expressions for the generating functions involved, here we show how to get the required information for the asymptotic estimates with an indirect use of the existence of Puiseux expansions at singularities. We study, asymptotically and on average, the alphabetic size, the size of the [Formula: see text]-follow automaton and of the position automaton, as well as the ratio and the size of these expressions to standard regular expressions.

scholarly journals A Note on Alternating Sums

10.37236/1265 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Peter Kirschenhofer
Keyword(s):  
Generating Functions ◽  
Analysis Of Algorithms ◽  
Case Analysis ◽  
Contour Integration ◽  
Asymptotic Estimates ◽  
Average Case Analysis ◽  
Average Case ◽  
Complex Contour ◽  
Alternating Sums

We present some results on a certain type of alternating sums which frequently arise in connection with the average-case analysis of algorithms and data. Whereas the so-called Rice's method for treating such sums uses complex contour integration we perform manipulations of generating functions in order to get explicit results from which asymptotic estimates follow immediately.

Unified summation equations and their applications in tribology wear process

2013 ◽  
Vol 61 (2) ◽  
pp. 405-417 ◽  
Author(s):  
K. Wierzcholski
Keyword(s):  
Translation Operator ◽  
Wear Process ◽  
Recurrence Equations ◽  
New Form

Abstract This paper presents some applications of summation equations with regard to the calculation prognosis of micro-bearing wear parameters. Summation equations are presented in a new form of difference and recurrence equations where the unknown micro-bearing wear function occurs as the argument of the reciprocal unified operator of summation (UOS). In this paper the properties of the UOS and reciprocal UOS as well as the unitary translation operator (UTO) are defined and applied to a micro-bearing wear determination. The approach of both dual and interaction between summation and recurrence equations and micro-bearing wear determination, makes this paper unique.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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.

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