Search

In this paper, we propose a q-analogue of the number of permutations i(n, k) of length n having k inversions known by Mahonian numbers. We investigate useful properties and some combinatorial interpretations by lattice paths/partitions and tilings. Furthermore, we give two constructive proofs of the strong q-log-concavity of the q-Mahonian numbers in k and n, respectively. In particular for q = 1 , we obtain two constructive proofs of the log-concavity of the Mahonian numbers in k and n, respectively. [ABSTRACT FROM AUTHOR]

The enumeration of lattice paths is an important counting model in enumerative combinatorics. Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines, it has attracted much attention and is a hot research field. In this paper, we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions, steps, constrained conditions, the positions of starting and end points, and so on. (1) The progress of classical lattice path such as Dyck lattice is introduced. (2) A method to study the enumeration of lattice paths problem by generating function is introduced. (3) Some methods of studying the enumeration of lattice paths problem by matrix are introduced. (4) The family of lattice paths problem and some counting methods are introduced. (5) Some applications of family of lattice paths in symmetric function theory are introduced, and a related open problem is proposed. [ABSTRACT FROM AUTHOR]

We use results on Dyck words and lattice paths to derive a formula for the exact number of binary words of a given length with a given minimal abelian border length, tightening a bound on that number from Christodoulakis et al. (Discrete Applied Mathematics, 2014). We extend to any number of distinct abelian borders a result of Rampersad et al. (Developments in Language Theory, 2013) on the exact number of binary words of a given length with no abelian borders. We also generalize these results to partial words. [ABSTRACT FROM AUTHOR]

We begin our analysis with the study of two collections of lattice paths in the plane, denoted D[n,i,j]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {D}}_{[n,i,j]}$$\end{document} and P[n,i,j]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {P}}_{[n,i,j]}$$\end{document}. These paths consist of sequences of n steps, where each step allows movement in three directions: upward (with a maximum displacement of q units), rightward (exactly one unit), or downward (with a maximum displacement of p units). The paths start from the point (0, i) and end at the point (n, j). In the collection D[n,i,j]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {D}}_{[n,i,j]}$$\end{document}, it is a crucial constraint that paths never go below the x-axis, while in the collection P[n,i,j]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {P}}_{[n,i,j]}$$\end{document}, paths have no such restriction. We assign weights to each path in both collections and introduce weight polynomials and generating series for them. Our main results demonstrate that certain matrices of size q×p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q\times p$$\end{document} associated with these generating series can be expressed as matrix continued fractions. These results extend the notable contributions previously made by Flajolet (Discrete Math 32:125–161, 1980) and Viennot (Une Théorie Combinatoire des Polynômes Orthogonaux Généraux. University of Quebec at Montreal, Lecture Notes, 1983) in the scalar case p=q=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p=q=1$$\end{document}. The generating series can also be interpreted as resolvents of one-sided or two-sided difference operators of finite order. Additionally, we analyze a class of random banded matrices H, which have p+q+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p+q+1$$\end{document} diagonals with entries that are independent and bounded random variables. These random variables have identical distributions along diagonals. We investigate the asymptotic behavior of the expected values of eigenvalue moments for the principal n×n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\times n$$\end{document} truncation of H as n tends to infinity. [ABSTRACT FROM AUTHOR]

The number of down-steps between pairs of up-steps in kt-Dyck paths, a generalization of Dyck paths consisting of steps f(1; k); (1;-1)g such that the path stays (weakly) above the line y = -t, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between kt-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers. [ABSTRACT FROM AUTHOR]

We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set X ⊆ Z 2 , there are two naturally associated monoids: F X , the monoid of all X-walks/paths; and A X , the monoid of all endpoints of X-walks starting from the origin O. For each A ∈ A X , write π X (A) for the number of X-walks from O to A. Calculating the numbers π X (A) is a classical problem, leading to Fibonacci, Catalan, Motzkin, Delannoy and Schröder numbers, among many other well-studied sequences and arrays. Our main results give relationships between finiteness properties of the numbers π X (A) , geometrical properties of the step set X, algebraic properties of the monoid A X , and combinatorial properties of a certain bi-labelled digraph naturally associated to X. There is an intriguing divergence between the cases of finite and infinite step sets, and some constructions rely on highly non-trivial properties of real numbers. We also consider the case of walks constrained to stay within a given region of the plane. Several examples are considered throughout to highlight the sometimes-subtle nature of the theoretical results. [ABSTRACT FROM AUTHOR]

In the past twenty years, the enumeration of lattice walks with steps taken in a prescribed set S and confined to a given cone, especially the first quadrant of the plane, has been intensely studied. As a result, the generating functions of quadrant walks are now well-understood, provided the allowed steps are small, that is, S = {-1; 0; 1}². In particular, having small steps is crucial for the definition of a certain group of bi-rational transformations of the plane. It has been proved that this group is finite if and only if the corresponding generating function is D-finite (that is, it satisfies a linear differential equation with polynomial coefficients). This group is also the key to the uniform solution of 19 of the 23 small step models possessing a finite group. In contrast, almost nothing is known for walks with arbitrary steps. In this paper, we extend the definition of the group, or rather of the associated orbit, to this general case, and generalize the above uniform solution of small step models. When this approach works, it invariably yields a D-finite generating function. We apply it to many quadrant problems, including some infinite families. After developing the general theory, we consider the 13 110 two-dimensional models with steps in f􀀀2;􀀀1; 0; 1g2 having at least one 􀀀2 coordinate. We prove that only 240 of them have a finite orbit, and solve 231 of them with our method. The nine remaining models are the counterparts of the four models of the small step case that resist the uniform solution method (and which are known to have an algebraic generating function). We conjecture D-finiteness for their generating functions, but only two of them are likely to be algebraic. We also prove non-D-finiteness for the 12 870 models with an infinite orbit, except for 16 of them. [ABSTRACT FROM AUTHOR]

We consider two related models for the propagation of a curvature sensitive interface in a time independent random medium. In both cases we suppose that the medium contains obstacles that act on the propagation of the interface with an inhibitory or an acceleratory force. We show that the interface remains bounded for all times even when a small constant external driving force is applied. This phenomenon has already been known when only inhibitory obstacles are present. In this work we extend this result to the case of - for example - a random medium of random zero mean forcing. The first model we study is discrete with a random forcing on each lattice site. In this case we construct a supersolution employing a local path optimization procedure. In the second, continuous, model we consider a random heterogenous medium consisting of localized small obstacles of random sign. To construct a stationary supersolution here, we need to pass through sufficiently many blocking obstacles while avoiding any obstacles of the other sign. This is done by employing a custom percolation argument. [ABSTRACT FROM AUTHOR]

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers. [ABSTRACT FROM AUTHOR]

Let λ be a partition with no more than n parts. Let β be a weakly increasing n-tuple with entries from f1; :::; ng. The flagged Schur function in the variables x1; :::; xn that is indexed by λ and β has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape λ whose values are row-wise bounded by the entries of β. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by λ and β; this could be done since the pair (λ; β) satisfied their "nonpermutable" condition for the sequence of terminals of an n-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that β be weakly increasing. Then for each λ we give a condition on the entries of β for the pair (λ; β) to be nonpermutable that is both necessary and sufficient. When the parts of λ are not distinct there will be multiple row bound n-tuples β that will produce the same set of tableaux. We accordingly group the bounding β into equivalence classes and identify the most efficient β in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by n and λ are enumerated by the number of these efficient n-tuples. We called these counts "parabolic Catalan numbers". It is noted that the GL(n) Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants. [ABSTRACT FROM AUTHOR]

In this paper a natural question which arise to study the graphical aspect of split (n + t) -color partitions, is answered by introducing a new class of lattice paths, called split lattice paths. A direct bijection between split (n + t) -color partitions and split lattice paths is proved. This new combinatorial object is applied to give new combinatorial interpretations of two basic functions of Gordon-McIntosh. Some generalized q-series are also discussed. We further explore these paths by providing combinatorial interpretations of some Rogers-Ramanujan type identities which reveal their rich structure and great potential for further research. [ABSTRACT FROM AUTHOR]

The kinetics of molecular hydrogen reactions at the Si (100) surface has been studied by simulation to extract the physics underlying two unexpected experimental observations: apparently first-order desorption kinetics and an increase in sticking probability with hydrogen coverage. At a partially H-terminated Si(100) surface, each Si dimer assumes an unoccupied dimer (UOD), singly occupied dimer (SOD), or doubly occupied dimer (DOD) structure. In our hydrogen reaction model based on an inter-dimer mechanism, a site consisting of an adjacent pair of a DOD and a UOD (DOD/UOD) is a key component for the desorption and adsorption kinetics of hydrogen at the Si(100) surface. To simulate reaction kinetics of both reactions, DU (D: DOD, U: UOD) and SS (S: SOD) pathways are proposed: DU pathway claims that the adsorption as well as desorption of hydrogen takes place at common sites having a cis-configured SOD/SOD pair that is transformed transiently from a DOD/UOD pair by H(D) diffusion. Thus the adsorption obeys the so-called 4H mechanism, but the desorption obeys the 2H mechanism. SS pathway claims that the adsorption occurs at sites having a UOD/UOD pair, and the desorption occurs at sites having a cis-configured SOD/SOD pair that is generated by diffusion of isolated SODs. To simulate temperature-programmed-desorption spectra and sticking probability vs coverage curves, thermo-statistics for a lattice-gas system characterized with parameters for hydrogen pairing and dimer clustering is used to evaluate equilibrium populations of DOD/UOD pairs and isolated SODs. The model simulation based on the above reaction model successfully reproduces all of the complicated, coverage dependent adsorption and desorption reactions of hydrogen at Si(100) surfaces. Specifically, at high coverage above 0.1 ML majority of the adsorption and desorption proceed along the DU pathway. Hence, it is suggested that the adsorption and desorption in the high coverage regime are not microscopically reversible. On the other hand, at low coverages below 0.1 ML, the simulation shows up that the majority of adsorption proceeds along the SS pathway, and the desorption by the DU pathway. Since both reactions obey the 2H mechanism, it is suggested that the desorption and adsorption in the low coverage regime are microscopically reversible. [ABSTRACT FROM AUTHOR]

Although the hammock networks were introduced more than sixty years ago, there is no general formula of the associated reliability polynomial. Using the full Hermite interpolation polynomial, we propose an approximation for the reliability polynomial of a hammock network of arbitrary size. In the second part of the paper, we provide combinatorial formulas for the first two non-zero coefficients of the reliability polynomial. [ABSTRACT FROM AUTHOR]

We use the tangent method to compute the arctic curve of the Twenty-Vertex (20V) model with particular domain wall boundary conditions for a wide set of integrable weights. To this end, we extend to the finite geometry of domain wall boundary conditions the standard connection between the bulk 20V and 6V models via the Kagome lattice ice model. This allows to express refined partition functions of the 20V model in terms of their 6V counterparts, leading to explicit parametric expressions for the various portions of its arctic curve. The latter displays a large variety of shapes depending on the weights and separates a central liquid phase from up to six different frozen phases. A number of numerical simulations are also presented, which highlight the arctic curve phenomenon and corroborate perfectly the analytic predictions of the tangent method. We finally compute the arctic curve of the Quarter Turn symmetric Holey Aztec Domino Tiling (QTHADT) model, a problem closely related to the 20V model and whose asymptotics may be analyzed via a similar tangent method approach. Again results for the QTHADT model are found to be in perfect agreement with our numerical simulations. [ABSTRACT FROM AUTHOR]

In this article we develop a vectorial kernel method—a powerful method which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton. We apply it for the enumeration of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern. We unify results from numerous articles concerning patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. This refines the study by Banderier and Flajolet from 2002 on enumeration and asymptotics of lattice paths: we extend here their results to pattern-avoiding walks/bridges/meanders/excursions. We show that the autocorrelation polynomial of this forbidden pattern, as introduced by Guibas and Odlyzko in 1981 in the context of rational languages, still plays a crucial role for our algebraic languages. En passant, our results give the enumeration of some classes of self-avoiding walks, and prove several conjectures from the On-Line Encyclopedia of Integer Sequences. Finally, we also give the trivariate generating function (length, final altitude, number of occurrences of the pattern p), and we prove that the number of occurrences is normally distributed and linear with respect to the length of the walk: this is what Flajolet and Sedgewick call an instance of Borges's theorem. [ABSTRACT FROM AUTHOR]

Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing these analogues can be applied to a wide variety of combinatorial sequences. As an application we develop a continuous analogue for the binomial distribution. [ABSTRACT FROM AUTHOR]

We are applying a new approach to interpret the generalized version of order 8 mock theta functions V 0 (q) and V 1 (q) in terms of n -color partitions. In literature, we find the interpretations of these functions in terms of split n -color partitions and signed partitions. In addition, we are interpreting the generalized versions of two mock theta functions of order 6 and two of order 5 in terms of n -color partitions. A mock theta function of order 3 is interpreted in terms of (n + 2) -color partitions. We further extend our results using lattice paths. [ABSTRACT FROM AUTHOR]

Abstract We evaluate a curious determinant, first mentioned by George Andrews in 1980 in the context of descending plane partitions. Our strategy is to combine the famous Desnanot-Jacobi-Dodgson identity with automated proof techniques. More precisely, we follow the holonomic ansatz that was proposed by Doron Zeilberger in 2007. We derive a compact and nice formula for Andrews's determinant, and use it to solve a challenge problem that we posed in a previous paper. By noting that Andrews's determinant is a special case of a two-parameter family of determinants, we find closed forms for several one-parameter subfamilies. The interest in these determinants arises because they count cyclically symmetric rhombus tilings of a hexagon with several triangular holes inside. [ABSTRACT FROM AUTHOR]

The Tangent Method of Colomo and Sportiello is applied to the study of the asymptotics of domino tilings of large Aztec rectangles, with some fixed distribution of defects along a boundary. The associated non-intersecting lattice path configurations are made of Schröder paths whose weights involve two parameters γ and q keeping track respectively of one particular type of step and of the area below the paths. We predict the arctic curve for an arbitrary distribution of defects, and illustrate our result with a number of examples involving different classes of boundary defects. [ABSTRACT FROM AUTHOR]

One of the most prominent problems in computational biology is to predict the natural conformation of a protein from its amino acid sequence. This paper focuses on the three-dimensional hydrophobic-polar (HP) lattice model of this problem. The modified elastic net (EN) algorithm is applied to solve this nonlinear programming hard problem. The lattice partition strategy and two local search methods (LS1 and LS2) are proposed to improve the performance of the modified EN algorithm. The computation and analysis of 12 HP standard benchmark instances are also involved in this paper. The results indicate that the hybrid of modified EN algorithm, lattice partition strategy, and local search methods has a greater tendency to form a globular state than genetic algorithm does. The results of noncompact model are more natural in comparison with that of compact model. [ABSTRACT FROM AUTHOR]

This article analyzes and explains the observed temperature dependence of the forward dark current of lattice matched In[sub 0.53]Ga[sub 0.47]As on InP diodes as a function of voltage. The experimental results show, at high temperatures, the characteristic current-voltage (I–V) curve corresponding to leakage, recombination, and diffusion currents, but at low temperatures an additional region is seen at high fields. We show that the onset of this region commences with high injection into the lower-doped base region. The high injection is shown by using simulation software to substantially alter the minority carrier concentration profile in the base, emitter and consequently the quasi-Fermi levels (QFL) at the base/window and the window/cap heterojunctions. We show that this QFL splitting and the associated electron “pile-up” (accumulation) at the window/emitter heterojunction leads to the observed pseudo-n=2 region of the current-voltage curve. We confirm this phenomenon by investigating the I–V–T characteristics of diodes with an InGaAsP quaternary layer (E[sub g]=1 eV) inserted between the InP window (E[sub g]=1.35 eV) and the InGaAs emitter (E[sub g]=0.72 eV) where it serves to reduce the barrier to injected electrons, thereby reducing the “pile-up.” We show, in this case that the high injection occurs at a higher voltage and lower temperature than for the ternary device, thereby confirming the role of the “accumulation” in the change of the I–V characteristics from n=1 to pseudo-n=2 in the ternary latticed matched device. This is an important phenomenon for consideration in thermophotovoltaic applications. We, also show that the activation energy at medium and high voltages corresponds to the InP/InGaAs conduction band offset at the window/emitter heterointerface. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Studies the variations in the lattice parameters of commercially available undoped and Silicon-doped Gallium Arsenide (GaAs) substrates using high resolution photoluminescence (PL) spectroscopy. Bound excitons of biaxially strained GaAs; Methodology of the study; Background about the GaAs grown on Si-doped substrates.

The transfer distance (TD) was introduced in the classification framework and studied in the context of phylogenetic tree matching. Recently, Lemoine et al. (Nature 556(7702):452–456, 2018. 10.1038/s41586-018-0043-0) showed that TD can be a powerful tool to assess the branch support on large phylogenies, thus providing a relevant alternative to Felsenstein's bootstrap. This distance allows a reference branch β in a reference tree T to be compared to a branch b from another tree T (typically a bootstrap tree), both on the same set of n taxa. The TD between these branches is the number of taxa that must be transferred from one side of b to the other in order to obtain β . By taking the minimum TD from β to all branches in T we define the transfer index, denoted by ϕ (β , T) , measuring the degree of agreement of T with β . Let us consider a reference branch β having p tips on its light side and define the transfer support (TS) as 1 - ϕ (β , T) / (p - 1) . Lemoine et al. (2018) used computer simulations to show that the TS defined in this manner is close to 0 for random "bootstrap" trees. In this paper, we demonstrate that result mathematically: when T is randomly drawn, TS converges in probability to 0 when n tends to ∞ . Moreover, we fully characterize the distribution of ϕ (β , T) on caterpillar trees, indicating that the convergence is fast, and that even when n is small, moderate levels of branch support cannot appear by chance. [ABSTRACT FROM AUTHOR]

Inversion sequences are in natural bijection with permutations and have surprising connections with lecture hall polytopes and partitions. Recently, Martinez and Savage carried out the systematic study of inversion sequences avoiding triples of relations. They established many connections with known integer sequences and highlighted several interesting conjectures, some of which have already been solved. In this paper, we address the remaining enumeration conjectures posed by them, leaving only those related to the OEIS sequence A098746 open. [ABSTRACT FROM AUTHOR]

Abstract Borel's triangle is an array of integers closely related to the classical Catalan numbers. In this paper we study combinatorial statistics counted by Borel's triangle. We present various combinatorial interpretations of Borel's triangle in terms of lattice paths, binary trees, and pattern avoiding permutations and matchings, and derive a functional equation that is useful in analyzing the involved structures. [ABSTRACT FROM AUTHOR]

Abstract The number of Borel orbits in the symmetric space S L n ∕ S (G L p × G L q) is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored. [ABSTRACT FROM AUTHOR]

For fixed integers n(= 0) and μ, the number of ways in which a moving particle taking a horizontal step with probability p and a vertical step with probability q, touches the line Y = n+μX for the first time, have been counted. The concept has been applied to obtain various probability distributions in independent and Markov dependent trials. [ABSTRACT FROM AUTHOR]

In this work we define a new set of integer partition, based on a lattice path in Z2 connecting the line x+y=n to the origin, which is determined by the two-line matrix representation given for different sets of partitions of n. The new partitions have only distinct odd parts with some particular restrictions. This process of getting new partitions, which has been called the Path Procedure, is applied to unrestricted partitions, partitions counted by the 1st and 2nd Rogers-Ramanujan Identities, and those generated by the Mock Theta Function T1∗(q)=∑n=0∞qn(n+1)(-q2,q2)n(q,q2)n+1. [ABSTRACT FROM AUTHOR]

Abstract Three summation formulae on the λ -extended Catalan numbers are established by means of hypergeometric series approach with one of them being provided a combinatorial proof through lattice path countings. [ABSTRACT FROM AUTHOR]

The dimensionality of self‐assembled nanostructures plays an essential role for their properties and applications. Herein, an understanding of the transition from weakly to strongly coupled layers in soft matter systems is provided involving in‐plane organized π‐conjugated rods. For this purpose, bolaamphiphilic triblock molecules consisting of a rigid biphenyl core, polar glycerol groups at the ends, and a branched (swallow‐tail) or linear alkyl or semiperfluoroalkyl chain in lateral position have been synthesized and investigated. Besides weakly coupled lamellar isotropic (LamIso), lamellar nematic (LamN) and sliding lamellar smectic phases (LamSm), a sequence of three distinct types of strongly coupled (correlated) lamellar smectic phases with either centered (c2mm) or non‐centered rectangular (p2mm) lattice and an intermediate oblique lattices (p2) were observed depending on chain length, chain branching and degree of chain fluorination. This new sequence is explained by the strengthening of the layer coupling and the competition between energetic packing constraints and the entropic contribution of either longitudinal or tangential fluctuations. This example of directed side chain engineering of small generic model compounds provides general clues for morphological design of two‐dimensional and three‐dimensionally coupled lamellar systems involving larger π‐conjugated molecular rods and molecular or supramolecular polymers, being of actual interest in organic electronics and nanotechnology. A transition from sliding to three distinct types of locked‐in lamellar phases with either centered (c2mm) or non‐centered rectangular (p2mm) lattice and an intermediate oblique lattices (p2) was observed by the strengthening of the layer coupling due to the competition between energetic packing constraints and the entropic contribution of either longitudinal or tangential fluctuations. [ABSTRACT FROM AUTHOR]

In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove that lattice path matroid polytopes are affinely equivalent to a family of distributive polytopes. As applications we obtain two new infinite families of matroids verifying a conjecture of De Loera et. al. and present an explicit formula of the Ehrhart polynomial for one of them. [ABSTRACT FROM AUTHOR]

Motivated by the relation N m ( C n ) = ( m n + 1 ) N m ( A n − 1 ) , holding for the m -generalized Catalan numbers of type A and C , the connection between dominant regions of the m -Shi arrangement of type A n − 1 and C n is investigated. More precisely, it is explicitly shown how m n + 1 copies of the set of dominant regions of the m -Shi arrangement of type A n − 1 , biject onto the set of type C n such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between m n + 1 copies of the set of m -Dyck paths of height n and the set of N − E lattice paths inside an n × m n rectangle is provided. [ABSTRACT FROM AUTHOR]

Studies the initial growth stages of the lattice-mismatched GaAs/InAs system by coaxial impact collision ion scattering spectroscopy. Background to the study; Results and discussion; Conclusion.

Presents a study that measured superconducting temperatures and magnetic fields, lattice parameters and chemical compositions for bulk layers of pure and alloyed niobium[sub3] tin which were made by the bronze process. Values of temperature of pure niobium[sub3] tin layers; Effect of small additions of titanium on the value of temperature; Experimental results.

Deals with a study which observed a flux-line lattice in a single crystal of Bi[sub2.15]Sr[sub1.95]CaCu[sub2]O[sub8+x] using small-angle neutron scattering methods. Background to the study; Experimental procedures; Results and discussion.

Architected 2D structures are of growing interest due to their unique mechanical and physical properties for applications in stretchable electronics, controllable phononic/photonic modulators, and switchable optical/electrical devices; however, the underpinning theory of understanding their elastic properties and enabling principles in search of emerging structures with welldefined arrangements and/or bonding connections of assembled elements has yet to be established. Here, we present two theoretical frameworks in mechanics--strain energy-based theory and displacement continuity-based theory--to predict the elastic properties of 2D structures and demonstrate their application in a search for novel architected 2D structures that are composed of heterogeneously arranged, arbitrarily shaped lattice cell structures with regulatory adjacent bonding connections of cells, referred to as heterogeneously architected 2D structures (HASs). By patterning lattice cell structures and tailoring their connections, the elastic properties of HASs can span a very broad range from nearly zero to beyond those of individual lattice cells by orders of magnitude. Interface indices that represent both the pattern arrangements of basic lattice cells and local bonding disconnections in HASs are also proposed and incorporated to intelligently design HASs with ondemand Young's modulus and geometric features. This study offers a theoretical foundation toward future architected structures by design with unprecedented properties and functions. [ABSTRACT FROM AUTHOR]

İn this paper, on one hand, we propose a new type of symmetric function to interpret the bis nomial coefficients and their analogs. On other hand, according to this function, we give an interpretation of these coefficients by lattice paths and tiling. Some identities of these coefficients are also established. This work is an extension of the results of Belbachir and Benmezai's "A q-analogue for bis nomial coefficients and generalized Fibonacci sequences". [ABSTRACT FROM AUTHOR]

For integers n , m with n ≥ 1 and 0 ≤ m ≤ n , an ( n , m ) -Dyck path is a lattice path in the integer lattice Z × Z using up steps ( 0 , 1 ) and down steps ( 1 , 0 ) that goes from the origin ( 0 , 0 ) to the point ( n , n ) and contains exactly m up steps below the line y = x . The classical Chung–Feller theorem says that the total number of ( n , m ) -Dyck path is independent of m and is equal to the n -th Catalan number C n = 1 n + 1 ( 2 n n ) . For any integer k with 1 ≤ k ≤ n , let p n , m , k be the total number of ( n , m ) -Dyck paths with k peaks. Ma and Yeh proved that p n , m , k = p n , n − m , n − k for 0 ≤ m ≤ n , and p n , m , k + p n , m , n − k = p n , m + 1 , k + p n , m + 1 , n − k for 1 ≤ m ≤ n − 2 . In this paper we give bijective proofs of these two results. Using our bijections, we also get refined enumeration results on the numbers p n , m , k and p n , m , k + p n , m , n − k according to the starting and ending steps. [ABSTRACT FROM AUTHOR]

Pt-modified β -NiAl is an alloy of special interest for bond coatings in the hot sections of the engines. Nevertheless, the site occupancy of Pt is still an open issue. In this work, we have constructed a Ni-Al-Pt interatomic potential and carried out the molecular dynamics simulations to reveal the atomic structures and displacement behaviours of Pt in B 2-NiAl. The results of the simulations not only show that the Pt atoms have a strong preference for the Ni site in B 2-NiAl but also suggest that an increasing Pt concentration leads to a more pronounced displacement of the Al atoms. Moreover, the simulations predict the occurrence of phase separation in B 2-NiAl when the Pt concentration reaches 20%, which is confirmed in the subsequent experiments by the formation of a β -NiAl+Al 2 Pt dual-phase structure. The present findings promote the understanding of the atomic structures in B 2-NiAl, which in turn may provide useful guidance for the fabrication of high-quality β -NiAl bond coatings. [ABSTRACT FROM AUTHOR]

For l≥1 and k≥2, we consider certain admissible sequences of k-1 lattice paths in a colored l×l square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of shape λ+l with l(λ)≤k, which is also the number of (k + 1)k··· 21-avoiding permutations in Sl. Finally, we apply this result to the representation theory of the affine Lie algebra sl(n) and show that this gives the multiplicity of certain maximal dominant weights in the irreducible highest weight sl(n) -module V(kλ0). [ABSTRACT FROM AUTHOR]

The Goulden–Jackson cluster method is a powerful tool for obtaining generating functions counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths – both with height bounded and unbounded – by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued fraction formulas. [ABSTRACT FROM AUTHOR]

For an infinite lower triangular matrix G = ( g n , k ) n , k ≥ 0 , we define the half of G to be the infinite lower triangular matrix H = ( h n , k ) n , k ≥ 0 such that h n , k = g 2 n − k , n for all n ≥ k ≥ 0 . In this paper, we will show that if G = ( g n , k ) n , k ≥ 0 is a Riordan array, then its half H = ( h n , k ) n , k ≥ 0 is also a Riordan array, and we obtain new combinatorial interpretations for some Riordan arrays in terms of weighted lattice paths. [ABSTRACT FROM AUTHOR]

The register function (or Horton–Strahler number) of a binary tree is a well-known combinatorial parameter. We study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree. In particular, the precise asymptotic behavior of the number of certain substructures (“branches”) that occur when reducing a tree repeatedly is determined. In the same manner we introduce a reduction for simple two-dimensional lattice paths from which a complexity measure similar to the register function can be derived. We analyze this quantity, as well as the (cumulative) size of an (iteratively) reduced lattice path asymptotically. [ABSTRACT FROM AUTHOR]

Using density-functional total energy calculations, we investigated N interstitial migration in GaN. Two migration paths were considered. The first path confines motion to a single c-plane of the lattice, while the second path involves movement both perpendicular and parallel to the c-axis. The latter path has a lower barrier for the positive charge states and will be the dominant mechanism for migration of the N interstitial in p-type GaN. The calculated barriers are 1.79, 2.12, and 1.98 eV for the +1, +2, and +3 charge states. These barriers are consistent with recent experimental results and indicate that interstitials will be mobile at typical processing temperatures. [ABSTRACT FROM AUTHOR]

A dispersed Dyck path (DDP) of length is a lattice path on from to in which the following steps are allowed: 'up' ; 'down' ; and 'right' . An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length , #A191386 in Sloane's OEIS. Previously, only implicit generating function relations and asymptotics were known. [ABSTRACT FROM AUTHOR]

We show that problems of existence and characterization of wavelets for non-expanding dilations are intimately connected with the geometry of numbers; more specifically, with a bound on the number of lattice points in balls dilated by the powers of a dilation matrix A ∈ GL(n,ℝ). This connection is not visible for the well-studied class of expanding dilations since the desired lattice counting estimate holds automatically. We show that the lattice counting estimate holds for all dilations A with ∣detA∣ ≠ 1 and for almost every lattice Γ with respect to the invariant probability measure on the set of lattices. As a consequence, we deduce the existence of minimally supported frequency (MSF) wavelets associated with such dilations for almost every choice of a lattice. Likewise, we show that MSF wavelets exist for all lattices and almost every choice of a dilation A with respect to the Haar measure on GL(n,ℝ). [ABSTRACT FROM AUTHOR]

We consider a generalised version of Motzkin paths, where horizontal steps have length ℓ , with ℓ being a fixed positive integer. We first give the general functional equation for the area-width generating function of this model. Using a heuristic ansatz, we then derive the area-width scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schröder paths, before generalising our approach to arbitrary ℓ . We then rigorously derive the tricritical scaling of Schröder paths by applying the generalised method of steepest descents to the known exact solution for their area-width generating function. Our results show that for Dyck and Schröder paths, the heuristic scaling ansatz reproduces the rigorous results. [ABSTRACT FROM AUTHOR]

The article focus on solving quarter-plane lattice walks with interactions via the kernel method. Topics include number of n-step paths that start at point which visit vertices on the horizontal boundary; and how interactions will affect the solubility of quarter-plane lattice walk models and the properties of the generating functions Q(x, y, t).

Two statistics with respect to “upper-corners” and “lower-corners” are introduced for lattice paths. The corresponding refined generating functions are shown to be closely related to the q -ballot polynomials that extend the well-known Narayana polynomials and Catalan numbers. [ABSTRACT FROM AUTHOR]

In this paper we present a combinatorial generalization of the fact that the number of plane partitions that fit in a 2a × b × b box is equal to the number of such plane partitions that are symmetric, times the number of such plane partitions for which the transpose is the same as the complement. We use the equivalent phrasing of this identity in terms of symmetry classes of lozenge tilings of a hexagon on the triangular lattice. Our generalization consists of allowing the hexagon to have certain symmetrically placed holes along its horizontal symmetry axis. The special case when there are no holes can be viewed as a new, simpler proof of the enumeration of symmetric plane partitions [ABSTRACT FROM AUTHOR]