ON THE LATTICE PATH METHOD IN CONVOLUTION-TYPE COMBINATORIAL IDENTITIES(Ⅱ)—THE WEIGHTED COUNTING FUNCTION METHOD ON LATTICE PATHS

An independent method for paper [10] is presented. Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multinomial coefficient in the case of unrestricted paths. Convolutions for path counts are investigated, which yields some Vandcrmondc-type identities for multinomial and q-multinomial coefficients.

Greedy Lattice Paths with General Weights

Let{X_(v):v∈Z^(d)}be i.i.d.random variables.Let S(π)=Σ_(v∈π)X_(v)be the weight of a self-avoiding lattice pathπ.Let M_(n)=max{S(π):πhas length n and starts from the origin}.We are interested in the asymptotics of Mn as n→∞.This model is closely related to the first passage percolation when the weights{X_(v):v∈Z^(d)}are non-positive and it is closely related to the last passage percolation when the weights{X_(v):v∈Z^(d)}are non-negative.For general weights,this model could be viewed as an interpolation between first passage models and last passage models.Besides,this model is also closely related to a variant of the position of right-most particles of branching random walks.Under the two assumptions that∃α>0,E(X_(0)^(+))^(d)(log^(+)X_(0)^(+))^(d+α)<+∞and that E[X_(0)^(−)]<+∞,we prove that there exists a finite real number M such that Mn/n converges to a deterministic constant M in L^(1)as n tends to infinity.And under the stronger assumptions that∃α>0,E(X_(0)^(+))^(d)(log^(+)X_(0)^(+))^(d+α)<+∞and that E[(X_(0)^(−))^(4)]<+∞,we prove that M_(n)/n converges to the same constant M almost surely as n tends to infinity.

A Note on Lattice Paths with Diagonal Steps in Three-Dimensional Space

Jr. Stocks[4] discussed lattice paths from （0, 0, 0） to （n, n, n） with diagonal steps under some restrictions. In this note, we give simpler formulas for the main results in [4], andextend them to a general case.

The research and progress of the enumeration of lattice paths

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.

Two Bijections on Weighted Motzkin Paths

In this paper, we provide a bijection between the set of underdiagonal lattice paths of length n and the set of(2, 2)-Motzkin paths of length n. Besides, we generalize the bijection of Shapiro and Wang(Shapiro L W, Wang C J. A bijection between 3-Motzkin paths and Schr¨oder paths with no peak at odd height. J. Integer Seq., 2009, 12: Article 09.3.2.) to a bijection between k-Motzkin paths and(k-2)-Schr¨oder paths with no horizontal step at even height. It is interesting that the second bijection is a generalization of the well-known bijection between Dyck paths and 2-Motzkin paths.

Variable dimensional state space based global path planning for mobile robot

A variable dimensional state space（VDSS） has been proposed to improve the re-planning time when the robotic systems operate in large unknown environments.VDSS is constructed by uniforming lattice state space and grid state space.In VDSS,the lattice state space is only used to construct search space in the local area which is a small circle area near the robot,and grid state space elsewhere.We have tested VDSS with up to 80 indoor and outdoor maps in simulation and on segbot robot platform.Through the simulation and segbot robot experiments,it shows that exploring on VDSS is significantly faster than exploring on lattice state space by Anytime Dynamic A＊（AD＊） planner and VDSS is feasible to be used on robotic systems.