scholarly journals Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption

10.37236/1637 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
E. J. Janse van Rensburg ◽  
A. Rechnitzer
Keyword(s):  
Critical Point ◽  
Generating Functions ◽  
Asymptotic Expression ◽  
Path Model ◽  
Motzkin Path ◽  
Dyck Paths ◽  
Asymptotic Expressions ◽  
Directed Walks ◽  
Open Question ◽  
Vertex Colouring

In a previous work [26], by considering paths that are partially weighted, the generating function of Dyck paths was shown to possess a type of symmetry, called an exchange relation, derived from the exchange of a portion of the path between weighted and unweighted halves. This relation is particularly useful in solving for the generating functions of certain models of vertex-coloured Dyck paths; this is a directed model of copolymer adsorption, and in a particular case it is possible to find an asymptotic expression for the adsorption critical point of the model as a function of the colouring. In this paper we examine Motzkin path and partially directed walk models of the same adsorbing directed copolymer problem. These problems are an interesting generalisation of previous results since the colouring can be of either the edges, or the vertices, of the paths. We are able to find asymptotic expressions for the adsorption critical point in the Motzkin path model for both edge and vertex colourings, and for the partially directed walk only for edge colourings. The vertex colouring problem in partially directed walks seems to be beyond the scope of the methods of this paper, and remains an open question. In both these cases we first find exchange relations for the generating functions, and use those to find the asymptotic expression for the adsorption critical point.

scholarly journals Minimal and maximal plateau lengths in Motzkin paths

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Helmut Prodinger ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Gumbel Distribution ◽  
Minimal Length ◽  
Analytic Method ◽  
Horizontal Segment ◽  
Motzkin Path ◽  
Dyck Paths ◽  
Substitution Process ◽  
International Audience ◽  
Motzkin Paths

International audience The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters <i> maximal plateau length, horizontal segment </i>and <i>maximal horizontal segment </i>as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.

scholarly journals The location of the first maximum in the first sojourn of a Dyck path

2008 ◽  
Vol Vol. 10 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Random Walks ◽  
Generating Functions ◽  
Analysis Of Algorithms ◽  
Dyck Path ◽  
Dyck Paths ◽  
Asymptotic Expressions ◽  
International Audience

Analysis of Algorithms International audience For Dyck paths (nonnegative symmetric) random walks, the location of the first maximum within the first sojourn is studied. Generating functions and explicit resp. asymptotic expressions for the average are derived. Related parameters are also discussed.

scholarly journals Generating functions for a lattice path model introduced by Deutsch

2021 ◽  
Vol 9 (1) ◽  
pp. 217-225
Author(s):  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Path Model ◽  
Lattice Path ◽  
Dyck Paths

Abstract The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size −3, −5, −7, . . . . For such paths, we find the generating functions of them, according to length, ending at level i, both, when considering them from left to right and from right to left. The generating functions are intrinsically cubic, and thus (for i = 0) in bijection to various objects, like even trees, ternary trees, etc.

scholarly journals Local and global extrema for functions of several variables

1980 ◽  
Vol 29 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Bruce Calvert ◽  
M. K. Vamanamurthy
Keyword(s):  
Critical Point ◽  
Local Minimum ◽  
Global Minimum ◽  
Sufficient Conditions ◽  
Subject Classification ◽  
General Function ◽  
Several Variables ◽  
Open Question

AbstractLet p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

scholarly journals A General Theory of Wilf-Equivalence for Catalan Structures

10.37236/5629 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
General Theory ◽  
Equivalence Relation ◽  
Generating Functions ◽  
Asymptotic Estimate ◽  
Equivalence Classes ◽  
Catalan Number ◽  
Combinatorial Structures ◽  
Dyck Paths ◽  
Significant Interest ◽  
Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

The Enumeration of Maps on the Torus and the Projective Plane

1988 ◽  
Vol 31 (3) ◽  
pp. 257-271 ◽  
Author(s):  
E. A. Bender ◽  
E. R. Canfield ◽  
R. W. Robinson
Keyword(s):  
Projective Plane ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Embedded Graphs ◽  
Asymptotic Expressions ◽  
Explicit Expressions

AbstractThe enumeration of rooted maps (embedded graphs), by number of edges, on the torus and projective plane, is studied. Explicit expressions for the generating functions are obtained. From these are derived asymptotic expressions and recurrence relations. Numerical tables for the numbers with up to 20 edges are presented.

scholarly journals Functional advantages of Lévy walks emerging near a critical point

2020 ◽  
Vol 117 (39) ◽  
pp. 24336-24344 ◽  
Author(s):  
Masato S. Abe
Keyword(s):  
Critical Point ◽  
Dynamic Range ◽  
Movement Trajectories ◽  
Lévy Walks ◽  
Optimal Dynamic ◽  
Freely Moving ◽  
High Flexibility ◽  
Open Question ◽  
The Brain ◽  
Power Law Distributions

A special class of random walks, so-called Lévy walks, has been observed in a variety of organisms ranging from cells, insects, fishes, and birds to mammals, including humans. Although their prevalence is considered to be a consequence of natural selection for higher search efficiency, some findings suggest that Lévy walks might also be epiphenomena that arise from interactions with the environment. Therefore, why they are common in biological movements remains an open question. Based on some evidence that Lévy walks are spontaneously generated in the brain and the fact that power-law distributions in Lévy walks can emerge at a critical point, we hypothesized that the advantages of Lévy walks might be enhanced by criticality. However, the functional advantages of Lévy walks are poorly understood. Here, we modeled nonlinear systems for the generation of locomotion and showed that Lévy walks emerging near a critical point had optimal dynamic ranges for coding information. This discovery suggested that Lévy walks could change movement trajectories based on the magnitude of environmental stimuli. We then showed that the high flexibility of Lévy walks enabled switching exploitation/exploration based on the nature of external cues. Finally, we analyzed the movement trajectories of freely moving Drosophila larvae and showed empirically that the Lévy walks may emerge near a critical point and have large dynamic range and high flexibility. Our results suggest that the commonly observed Lévy walks emerge near a critical point and could be explained on the basis of these functional advantages.

ANISOTROPIC STEP, SURFACE CONTACT, AND AREA WEIGHTED DIRECTED WALKS ON THE TRIANGULAR LATTICE

2002 ◽  
Vol 16 (09) ◽  
pp. 1269-1299 ◽  
Author(s):  
A. C. OPPENHEIM ◽  
R. BRAK ◽  
A. L. OWCZAREK
Keyword(s):  
Triangular Lattice ◽  
Generating Functions ◽  
Continued Fractions ◽  
Functional Equations ◽  
Square Lattice ◽  
Combinatorial Objects ◽  
Special Cases ◽  
Explicit Formulae ◽  
Directed Walks ◽  
Marked Area

We present results for the generating functions of single fully-directed walks on the triangular lattice, enumerated according to each type of step and weighted proportional to the area between the walk and the surface of a half-plane (wall), and the number of contacts made with the wall. We also give explicit formulae for total area generating functions, that is when the area is summed over all configurations with a given perimeter, and the generating function of the moments of heights above the wall (the first of which is the total area). These results generalise and summarise nearly all known results on the square lattice: all the square lattice results can be obtaining by setting one of the step weights to zero. Our results also contain as special cases those that already exist for the triangular lattice. In deriving some of the new results we utilise the Enumerating Combinatorial Objects (ECO) and marked area methods of combinatorics for obtaining functional equations in the most general cases. In several cases we give our results both in terms of ratios of infinite q-series and as continued fractions.

scholarly journals Variations on Descents and Inversions in Permutations

10.37236/856 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Denis Chebikin
Keyword(s):  
Generating Functions ◽  
Euler Number ◽  
Weighted Sum ◽  
Eulerian Polynomials ◽  
Joint Distributions ◽  
Dyck Paths ◽  
Sigma 1 ◽  
Descent Set ◽  
New Statistics ◽  
Factor Set

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation $\sigma = \sigma_1\sigma_2\cdots\sigma_n$ defined as the set of indices $i$ such that either $i$ is odd and $\sigma_i > \sigma_{i+1}$, or $i$ is even and $\sigma_i < \sigma_{i+1}$. We show that this statistic is equidistributed with the odd $3$-factor set statistic on permutations $\tilde{\sigma} = \sigma_1\sigma_2\cdots\sigma_{n+1}$ with $\sigma_1=1$, defined to be the set of indices $i$ such that the triple $\sigma_i \sigma_{i+1} \sigma_{i+2}$ forms an odd permutation of size $3$. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same, establishing a connection to two classical Mahonian statistics, maj and stat, along the way. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials $\sum_{\sigma\in\mathcal{S}_n} t^{{\rm des}(\sigma)+1}$ using alternating descents. For the alternating descent set statistic, we define the generating polynomial in two non-commutative variables by analogy with the $ab$-index of the Boolean algebra $B_n$, providing a link to permutations without consecutive descents. By looking at the number of alternating inversions, which we define in the paper, in alternating (down-up) permutations, we obtain a new $q$-analog of the Euler number $E_n$ and show how it emerges in a $q$-analog of an identity expressing $E_n$ as a weighted sum of Dyck paths.

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