scholarly journals Modular, $k$-Noncrossing Diagrams

10.37236/348 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Christian M. Reidys ◽  
Rita R. Wang ◽  
Albus Y. Y. Zhao
Keyword(s):  
Asymptotic Formula ◽  
Generating Function ◽  
Exact Enumeration ◽  
Tertiary Structures ◽  
Basic Features

In this paper we compute the generating function of modular, $k$-noncrossing diagrams. A $k$-noncrossing diagram is called modular if it does not contain any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA tertiary structures and their properties reflect basic features of these bio-molecules. The particular case of modular noncrossing diagrams has been extensively studied. Let ${Q}_k(n)$ denote the number of modular $k$-noncrossing diagrams over $n$ vertices. We derive exact enumeration results as well as the asymptotic formula ${Q}_k(n)\sim c_k n^{-(k-1)^2-{k-1\over2}}\gamma_{k}^{-n}$ for $k=3, \ldots, 9$ and derive a new proof of the formula ${Q}_2(n)\sim 1.4848\, n^{-3/2}\,1.8489^{n}$ (Hofacker et al. 1998).

Asymptotic Expansions II

1957 ◽  
Vol 9 ◽  
pp. 194-209 ◽  
Author(s):  
Leo Moser ◽  
Max Wyman
Keyword(s):  
Asymptotic Formula ◽  
Generating Function ◽  
Asymptotic Expansions ◽  
Image Position

In a previous paper (1) the authors considered the problem of finding an asymptotic formula for numbers or functions Bn,m whose generating function is of the form(1.1),where Pm(x) is a polynomial of degree m in x given by(1.2), am≠0.The above-mentioned paper contained the restriction that ak ≥ 0.

scholarly journals ON THE DISTRIBUTION OF THE RANK STATISTIC FOR STRONGLY CONCAVE COMPOSITIONS

2019 ◽  
Vol 100 (2) ◽  
pp. 230-238 ◽  
Author(s):  
NIAN HONG ZHOU
Keyword(s):  
Number Theory ◽  
Asymptotic Formula ◽  
Generating Function ◽  
Rank Statistic ◽  
Integer Partition

A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

scholarly journals Distinct Parts Partitions without Sequences

10.37236/4970 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Karl Mahlburg ◽  
Karthik Nataraj
Keyword(s):  
Asymptotic Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Series Representation ◽  
Double Series ◽  
Asymptotic Formulas ◽  
Similar Investigation ◽  
Combinatorial Properties ◽  
Consecutive Integers

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the generating function, an asymptotic formula for the enumeration function, and several combinatorial inequalities.

scholarly journals On an open problem of Green and Losonczy: exact enumeration of freely braided permutations

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Toufik Mansour
Keyword(s):  
Representation Theory ◽  
Generating Function ◽  
Coxeter Group ◽  
Upper Bound ◽  
Open Problem ◽  
Special Class ◽  
Exact Enumeration ◽  
Restricted Permutations ◽  
International Audience ◽  
Freely Braided

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

scholarly journals $321$-Polygon-Avoiding Permutations and Chebyshev Polynomials

10.37236/1677 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Toufik Mansour ◽  
Zvezdelina Stankova
Keyword(s):  
Symmetric Group ◽  
Explicit Expression ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Schubert Varieties ◽  
Exact Enumeration ◽  
Linear Recurrences ◽  
Singular Loci ◽  
Constant Coefficients

A $321$-$k$-gon-avoiding permutation $\pi$ avoids $321$ and the following four patterns: $$k(k+2)(k+3)\cdots(2k-1)1(2k)23\cdots(k-1)(k+1),$$ $$k(k+2)(k+3)\cdots(2k-1)(2k)12\cdots(k-1)(k+1),$$ $$(k+1)(k+2)(k+3)\cdots(2k-1)1(2k)23\cdots k,$$ $$(k+1)(k+2)(k+3)\cdots(2k-1)(2k)123\cdots k.$$ The $321$-$4$-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincaré polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases $k=2,3,4$. In this paper, we extend these results by finding an explicit expression for the generating function for the number of $321$-$k$-gon-avoiding permutations on $n$ letters. The generating function is expressed via Chebyshev polynomials of the second kind.

scholarly journals On the Number of Permutations Admitting an m-th Root

10.37236/1620 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Nicolas Pouyanne
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Symmetric Group ◽  
Generating Function ◽  
Euler Function ◽  
Exponential Generating Function ◽  
Explicit Constant

Let $m$ be a positive integer, and $p_n(m)$ the proportion of permutations of the symmetric group $S_n$ that admit an $m$-th root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula $$p_n(m)\, \sim \, {{\pi _m}\over {n^{1-\varphi (m)/m}}},\;\; n\to \infty ,$$ where $\varphi$ is the Euler function and $\pi _m$ an explicit constant.

Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model

2007 ◽  
Vol 44 (02) ◽  
pp. 285-294 ◽  
Author(s):  
Qihe Tang
Keyword(s):  
Asymptotic Formula ◽  
Continuous Time ◽  
Heavy Tails ◽  
Tail Behavior ◽  
Renewal Model ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Tail Asymptotic

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

Torus Circumference and Thickness Parameters From a Linear DNA Size Series Suggest How Toruses Self-Assemble

1985 ◽  
Vol 43 ◽  
pp. 522-523
Author(s):  
George C. Ruben ◽  
Kenneth A. Marx
Keyword(s):  
Tertiary Structure ◽  
Dna Complexes ◽  
Tertiary Structures ◽  
Self Assembling ◽  
Double Stranded Dna ◽  
Calf Thymus ◽  
Dna Organization ◽  
Condensed Dna ◽  
Dna Size

Certain double stranded DNA bacteriophage and viruses are thought to have their DNA organized into large torus shaped structures. Morphologically, these poorly understood biological DNA tertiary structures resemble spermidine-condensed DNA complexes formed in vitro in the total absence of other macromolecules normally synthesized by the pathogens for the purpose of their own DNA packaging. Therefore, we have studied the tertiary structure of these self-assembling torus shaped spermidine- DNA complexes in a series of reports. Using freeze-etch, low Pt-C metal (10-15Å) replicas, we have visualized the microscopic DNA organization of both calf Thymus( CT) and linear 0X-174 RFII DNA toruses. In these structures DNA is circumferentially wound, continuously, around the torus into a semi-crystalline, hexagonal packed array of parallel DNA helix sections.

The self and schizophrenia: a cognitive approach

2003 ◽  
Vol 62 (1) ◽  
pp. 45-51 ◽  
Author(s):  
Marek Nieznanski
Keyword(s):  
Normal Subjects ◽  
Control Group ◽  
Cognitive Approach ◽  
Schizophrenic Patients ◽  
Self Schema ◽  
Persons With Schizophrenia ◽  
Basic Features ◽  
Trait Words ◽  
Patients Experience ◽  
Normal Controls

The aim of the study was to explore the basic features of self-schema in persons with schizophrenia. Thirty two schizophrenic patients and 32 normal controls were asked to select personality trait words from a check-list that described themselves, themselves as they were five years ago, and what most people are like. Compared with the control group, participants from the experimental group chose significantly more adjectives that were common to descriptions of self and others, and significantly less that were common to self and past-self descriptions. These results suggest that schizophrenic patients experience their personality as changing over time much more than do healthy subjects. Moreover, their self-representation seems to be less differentiated from others-representation and less clearly defined than in normal subjects.

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