scholarly journals Asymptotics of the Stirling numbers of the first kind revisited: A saddle point approach

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Guy Louchard
Keyword(s):  
Saddle Point ◽  
Generating Function ◽  
Central Region ◽  
Integral Formula ◽  
Stirling Numbers ◽  
Saddle Point Method ◽  
Asymptotic Results ◽  
Central Regions ◽  
Cauchy’S Integral Formula ◽  
International Audience

International audience Using the saddle point method, we obtain from the generating function of the Stirling numbers of the first kind [n j] and Cauchy's integral formula, asymptotic results in central and non-central regions. In the central region, we revisit the celebrated Goncharov theorem with more precision. In the region j = n - n(alpha); alpha > 1/2, we analyze the dependence of [n j] on alpha.

scholarly journals Asymptotics of the Stirling numbers of the second kind revisited

2013 ◽  
Vol 7 (2) ◽  
pp. 193-210 ◽  
Author(s):  
Guy Louchard
Keyword(s):  
Saddle Point ◽  
Generating Function ◽  
Central Region ◽  
Integral Formula ◽  
Stirling Numbers ◽  
Saddle Point Method ◽  
Asymptotic Results ◽  
Analytic Combinatorics ◽  
Central Regions ◽  
Cauchy’S Integral Formula

Using the Saddle point method and multiseries expansions, we obtain from the generating function of the Stirling numbers of the second kind {n / m} and Cauchy's integral formula, asymptotic results in central and non-central regions. In the central region, we revisit the celebrated Gaussian theorem with more precision. In the region m = n - na, 1 > a > 1/2, we analyze the dependence of {n / m} on a. An extension of some Moser and Wyman's result to full m range is also provided. This paper fits within the framework of Analytic Combinatorics.

scholarly journals Bipartite Random Graphs and Cuckoo Hashing

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Reinhard Kutzelnigg
Keyword(s):  
Saddle Point ◽  
Bipartite Graph ◽  
Random Graphs ◽  
Hash Table ◽  
Saddle Point Method ◽  
Asymptotic Results ◽  
Cuckoo Hashing ◽  
International Audience ◽  
Covering Tree ◽  
Number Of Cycles

International audience The aim of this paper is to extend the analysis of Cuckoo Hashing of Devroye and Morin in 2003. In particular we make several asymptotic results much more precise. We show, that the probability that the construction of a hash table succeeds, is asymptotically $1-c(\varepsilon)/m+O(1/m^2)$ for some explicit $c(\varepsilon)$, where $m$ denotes the size of each of the two tables, $n=m(1- \varepsilon)$ is the number of keys and $\varepsilon \in (0,1)$. The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph. We apply a double saddle point method to obtain asymptotic results covering tree sizes, the number of cycles and the probability that no complex component occurs.

scholarly journals The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Leonid Tolmatz
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Tail Asymptotics ◽  
Absolute Value ◽  
Exact Tail Asymptotics ◽  
International Audience ◽  
The Absolute

International audience The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

scholarly journals Asymptotics of Decomposable Combinatorial Structures of Alg-Log Type With Positive Log Exponent

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Zhicheng Gao ◽  
David Laferrière ◽  
Daniel Panario
Keyword(s):  
Generating Function ◽  
Asymptotic Results ◽  
Combinatorial Structures ◽  
International Audience ◽  
Decomposable Structures

International audience We consider the multiset construction of decomposable structures with component generating function $C(z)$ of alg-log type, $\textit{i.e.}$, $C(z) = (1-z)^{-\alpha} (\log \frac{1}{ 1-z})^{\beta}$. We provide asymptotic results for the number of labeled objects of size $n$ in the case when $\alpha$ is positive and $\beta$ is positive and in the case $\alpha = 0$ and $\beta \geq 2$. The case $0<-\alpha <1$ and any $\beta$ and the case $\alpha > 0$ and $\beta = 0$ have been treated in previous papers. Our results extend previous work of Wright.

scholarly journals Asymptotics for the partial fractions of the restricted partition generating function I

2016 ◽  
Vol 12 (06) ◽  
pp. 1421-1474 ◽  
Author(s):  
Cormac O’Sullivan
Keyword(s):  
Saddle Point ◽  
Generating Function ◽  
Point Method ◽  
Saddle Point Method ◽  
Partial Fraction ◽  
Restricted Partition ◽  
Partial Fraction Decomposition ◽  
Partial Fractions

The generating function for [Formula: see text], the number of partitions of [Formula: see text] into at most [Formula: see text] parts, may be written as a product of [Formula: see text] factors. We find the behavior of coefficients in the partial fraction decomposition of this product as [Formula: see text] by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher.

scholarly journals Asymptotics of Smallest Component Sizes in Decomposable Combinatorial Structures of Alg-Log Type

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Li Dong ◽  
Zhicheng Gao ◽  
Daniel Panario ◽  
Bruce Richmond
Keyword(s):  
Generating Function ◽  
Combinatorial Structure ◽  
Asymptotic Estimates ◽  
Asymptotic Results ◽  
Combinatorial Structures ◽  
Combinatorial Objects ◽  
International Audience ◽  
Restricted Pattern

International audience A decomposable combinatorial structure consists of simpler objects called components which by thems elves cannot be further decomposed. We focus on the multi-set construction where the component generating function C(z) is of alg-log type, that is, C(z) behaves like c + d(1 -z/rho)(alpha) (ln1/1-z/rho)(beta) (1 + o(1)) when z is near the dominant singularity rho. We provide asymptotic results about the size of thes mallest components in random combinatorial structures for the cases 0 < alpha < 1 and any beta, and alpha < 0 and beta=0. The particular case alpha=0 and beta=1, the so-called exp-log class, has been treated in previous papers. We also provide similar asymptotic estimates for combinatorial objects with a restricted pattern, that is, when part of its factorization patterns is known. We extend our results to include certain type of integers partitions. partitions

scholarly journals Joint String Complexity for Markov Sources

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Philippe Jacquet ◽  
Wojciech Szpankowski
Keyword(s):  
Saddle Point ◽  
Saddle Points ◽  
Singularity Analysis ◽  
Point Method ◽  
Saddle Point Method ◽  
Genome Sequences ◽  
Markov Source ◽  
International Audience ◽  
The Common ◽  
Markov Sources

International audience String complexity is defined as the cardinality of a set of all distinct words (factors) of a given string. For two strings, we define $\textit{joint string complexity}$ as the set of words that are common to both strings. We also relax this definition and introduce $\textit{joint semi-complexity}$ restricted to the common words appearing at least twice in both strings. String complexity finds a number of applications from capturing the richness of a language to finding similarities between two genome sequences. In this paper we analyze joint complexity and joint semi-complexity when both strings are generated by a Markov source. The problem turns out to be quite challenging requiring subtle singularity analysis and saddle point method over infinity many saddle points leading to novel oscillatory phenomena with single and double periodicities.

scholarly journals A further generalization of the Catalan numbers and its explicit formula and integral representation

2021 ◽  
Author(s):  
Wen-Hui Li ◽  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Number Theory ◽  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Complex Analysis ◽  
Integral Formula ◽  
Catalan Numbers ◽  
Combinatorial Number Theory ◽  
Integral Formulas ◽  
Cauchy’S Integral Formula

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy’s integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

scholarly journals A further generalization of the Catalan numbers and its explicit formula and integral representation

2020 ◽  
Author(s):  
Wen-Hui Li ◽  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Number Theory ◽  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Complex Analysis ◽  
Integral Formula ◽  
Catalan Numbers ◽  
Combinatorial Number Theory ◽  
Integral Formulas ◽  
Cauchy’S Integral Formula

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy's integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

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