Lattice Paths Research Articles

Lattice paths and the Prouhet-Thue-Morse sequence

Using Flajolet's combinatorial theory of continued fractions and axial symmetry of lattice paths, we show that the enumeration of certain lattice paths modulo two yields the ubiquitous Prouhet-Thue-Morse sequence. This answers an open problem of Berstel et al.

Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises

Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in [Formula: see text] starting at the origin made of steps [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], where two down-steps cannot be consecutive. We enumerate MAP and their prefixes avoiding peaks (respectively, valleys, respectively, double rise) according to the length, the type of the last step, and the height of its end-point. We express our results using Riordan arrays. Finally, we provide constructive bijections between these paths and restricted Dyck and Motzkin paths.

Efficient Representation of Lattice Path Matroids

Efficient Representation of Lattice Path Matroids

Off-diagonally symmetric domino tilings of the Aztec diamond of odd order

We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry property which states that the numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order 2n−1 are equal when the boundary defect is at the kth position and the (2n−k)th position on the boundary, respectively. This symmetry property proves a special case of a recent conjecture by Behrend, Fischer, and Koutschan.In the second direction, a Pfaffian formula is obtained for the number of “nearly” off-diagonally symmetric domino tilings of odd-order Aztec diamonds, where the entries of the Pfaffian satisfy a simple recurrence relation. The numbers of domino tilings mentioned in the above two directions do not seem to have a simple product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers. The proof of these results involves the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths. Finally, we propose conjectures concerning the log-concavity and asymptotic behavior of the number of off-diagonally symmetric domino tilings of odd-order Aztec diamonds.

Non-Noble-Metal-Based Electrocatalysts for Acidic Oxygen Evolution Reaction: Recent Progress, Challenges, and Perspectives.

The oxygen evolution reaction (OER) plays a pivotal role in diverse renewable energy storage and conversion technologies, including water electrolysis, electrochemical CO2 reduction, nitrogen fixation, and metal-air batteries. Among various water electrolysis techniques, proton exchange membrane (PEM)-based water electrolysis devices offer numerous advantages, including high current densities, exceptional chemical stability, excellent proton conductivity, and high-purity H2. Nevertheless, the prohibitive cost associated with Ir/Ru-based OER electrocatalysts poses a significant barrier to the broad-scale application of PEM-based water splitting. Consequently, it is crucial to advance the development of non-noble metal OER catalysis substance with high acid-activity and stability, thereby fostering their widespread integration into PEM water electrolyzers (PEMWEs). In this review, a comprehensive analysis of the acidic OER mechanism, encompassing the adsorbate evolution mechanism (AEM), lattice oxygen mechanism (LOM) and oxide path mechanism (OPM) is offered. Subsequently, a systematic summary of recently reported noble-metal-free catalysts including transition metal-based, carbon-based and other types of catalysts is provided. Additionally, a comprehensive compilation of in situ/operando characterization techniques is provided, serving as invaluable tools for furnishing experimental evidence to comprehend the catalytic mechanism. Finally, the present challenges and future research directions concerning precious-metal-free acidic OER are comprehensively summarized and discussed in this review.

Greedy Lattice Paths with General Weights

Greedy Lattice Paths with General Weights

Lattice paths in corridors and cyclic corridors

Lattice paths in corridors and cyclic corridors

Combinatorial Generation Algorithms for Directed Lattice Paths

Combinatorial Generation Algorithms for Directed Lattice Paths

Lattice Path Matroids and Quotients

We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that—contrary to arbitrary positroid flag matroids—they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.

Matrix Continued Fractions Associated with Lattice Paths, Resolvents of Difference Operators, and Random Polynomials

Matrix Continued Fractions Associated with Lattice Paths, Resolvents of Difference Operators, and Random Polynomials

Sintering Behavior and Mechanical Properties of Dispersed-Reinforced Fe-TaC Composites Produced by High Energy Ball Milling

In this study, a high-melting point element, tantalum carbide (TaC) was added to improve the mechanical properties and the oxidation/corrosion resistance of Fe. The Fe-TaC composites were synthesized by high energy ball milling for the mechanical alloying of the non-equilibrium phase (Fe-Ta) and the homogeneous dispersion of TaC. Fe-TaC composite samples were fabricated using 5, 10, and 20 wt.% TaC. The ductile particles (Fe) got harden and the brittle particles (TaC) were uniformly dispersed, while facilitating short-range diffusion in the ductile matrix by the high energy ball milling method. Spark plasma sintering was performed at a sintering temperature of 850 oC and pressure of 60 MPa. As the TaC contents increased, the sintering exponential (m) increased. A higher ‘m’ value indicates a lower magnitude of shrinkage, by decreasing the lattice and grain boundary (G/B) diffusion path between the pores and particles. The hardness increased from 128.9±10.4 to 444.2±20.6 kg/mm<sup>2</sup> as the grain size decreased from 5.13 to 3.99 μm. This enhancement is attributed to the Hall-Petch relationship and dispersion strengthening effect. The mechanical properties of the sintered bodies were studied to evaluate how the different TaC content affect their characteristics. In addition, oxidation resistance increased with increasing TaC contents. It was considered that the local oxidation resistance based on the formation of an oxide layer of TaO and Fe<sub>2</sub>O<sub>3</sub>.

Riordan Arrays and Difference Equations of Subdiagonal Lattice Paths

Riordan Arrays and Difference Equations of Subdiagonal Lattice Paths

Logarithms of Catalan Generating Functions: A Combinatorial Approach

We analyze the combinatorics behind the operation of taking the logarithm of the generating function $G_k$ for $k^\text{th}$ generalized Catalan numbers. We provide combinatorial interpretations in terms of lattice paths and in terms of tree graphs. Using explicit bijections, we are able to recover known closed expressions for the coefficients of $\log G_k$ by purely combinatorial means of enumeration. The non-algebraic proof easily generalizes to higher powers $\log^a G_k$, $a\geq 2$.

Hamiltonian Intervals in the Lattice of Binary Paths

Let $\mathcal{P}_n$ be the set of all binary paths (i.e., lattice paths with upsteps $u = (1,1)$ and downsteps $d = (1,-1)$) of length $n$ endowed with the pointwise partial ordering (i.e., $P \le Q$ iff the lattice path $P$ lies weakly below $Q$) and let $G_n$ be its Hasse graph. For each path $P \in \mathcal{P}_n$, we denote by $I(P)$ the interval which contains the elements of $\mathcal{P}_n$ less than or equal to $P$, excluding the first two elements of $\mathcal{P}_n$, and by $G(P)$ the subgraph of $G_n$ induced by $I(P)$. In this paper, it is shown that $G(P)$ is Hamiltonian iff $P$ contains at least two peaks and $I(P)$ has equal number of elements with even and odd rank. The last condition is always true for paths ending with an upstep, whereas, for paths ending with a downstep, a simple characterization is given, based on the structure of the path.

Lattice Path Bicircular Matroids

Lattice path matroids and bicircular matroids are two well-known classes of transversal matroids. In the seminal work of Bonin and de Mier about structural properties of lattice path matroids, the authors claimed that lattice path matroids significantly differ from bicircular matroids. Recently, it was proved that all cosimple lattice path matroids have positive double circuits, while it was shown that there is a large class of cosimple bicircular matroids with no positive double circuits. These observations support Bonin and de Miers’ claim. Finally, Sivaraman and Slilaty suggested studying the intersection of lattice path matroids and bicircular matroids as a possibly interesting research topic. In this work, we exhibit the excluded bicircular matroids for the class of lattice path matroids, and we propose a characterization of the graph family whose bicircular matroids are lattice path matroids. As an application of this characterization, we propose a geometric description of 2-connected lattice path bicircular matroids.

Further Rogers-Ramanujan type identities for modified lattice paths

Recently, the authors introduced the modified lattice paths which generalize Agarwal-Bressoud weighted lattice paths. Using these new objects they interpreted combinatorially two basic series identities which led to two new combinatorial Rogers-Ramanujan type identities. In this paper we obtain three more Rogers-Ramanujan type identities for modified lattice paths. This also leads to three new 3-way combinatorial identities.

4-Dimensional Lattice Path Enumeration with Arbitrary Steps

<p>Consider a set of vectors, <em>L</em>, which consists of vectors whose coordinates are 0 or 1. We find explicit formulas that counts the number of lattice paths from origin to (<em>a</em>,<em>b</em>,<em>c</em>,<em>d</em>) for using vectors in {(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)} ∪ <em>L</em> for various choices of <em>L</em>. In some cases we also give the recursive formulas for the same problem. Next we determine the minimum number of vectors that must be used to reach (<em>a</em>,<em>b</em>,<em>c</em>,<em>d</em>), also called the minimum distance problem, for different sets of vectors.</p>

Trajectories of square lattice staircase polygons

The distribution of monomers in a coating of grafted and adsorbing polymers is modelled using a grafted staircase polygon in the square lattice. The adsorbing staircase polygon consists of a bottom and a top lattice path (branches) and the asymptotic probability density that vertices in the lattice are occupied by these paths is determined. This is a model consisting of two linear polymers grafted to a hard wall in a coating. The probability that either the bottom path, or the top path, of a staircase polygon passes through a lattice site with coordinates , for 0 < ϵ < 1 and δ ≥ 0, is determined asymptotically as n → ∞ . This gives the probability density of vertices in the staircase polygon in the scaling limit (as n → ∞ ): = The densities of other cases, grafted and adsorbed, and at the adsorption critical point (or special point), are also determined. In these cases the most likely and mean trajectories of the paths are determined. The results show that low density regions in the distribution induce entropic forces on test particles which confine them next to the hard wall, or between branches of the staircase polygon. These results give qualitative mechanisms for the stabilisation of a drug particle confined to a polymer coating in a drug delivery system such as a drug-eluding stent covered by a grafted polymer.

Lattice Paths, Vector Continued Fractions, and Resolvents of Banded Hessenberg Operators

Lattice Paths, Vector Continued Fractions, and Resolvents of Banded Hessenberg Operators

Hessenberg-Toeplitz matrix determinants with Schröder and Fine number entries

In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schröder and Fine number sequences. Algebraic proofs of these results can be given which make use of Trudi's formula and the generating function of the associated sequence of determinants. We also provide direct arguments of our results that utilize various counting techniques, among them sign-changing involutions, on combinatorial structures related to classes of lattice paths enumerated by the Schröder and Fine numbers. As a consequence of our results, we obtain some new formulas for the Schröder and Catalan numbers as well as for some additional sequences from the OEIS in terms of determinants of certain Hessenberg-Toeplitz matrices.