There is actually an one dimensional version of this problem. For each step of the lattice walk, we can move either east for one unit or west for one unit. The problem is that given a fixed $n$ steps of lattice walk with the last step has the farthest distance from the origin of the movement, how many ways are there for such movement? One dimensional is easy to solve as we can transform it to the two dimension dyke path which is an alternating explanation for the $Catalan$ $numbers$. However, is there way to solve the two dimensional cases? which means that there fours choices of movement which are either moving upward for one unit, downward for one unit, left for one unit or right for one unit, with a fixed number of steps, say $n$ and the last position of the lattice walk should have the farthest distance from the origin of the movement. The question is to find the number of such ways $a(n)$ in termes of $n $.

There is actually an one dimensional version of this problem. For each step of the lattice walk, we can move either east for one unit or west for one unit. The problem is that given a fixed $n$ steps of lattice walk with the last step has the farthest distance from the origin of the movement, the distance can be defined in the euclidean metric as the norm=$(x^2+y^2)^{1/2}$, and other norms can also be defined as max { $x,y$} or $x+y$ is also ok. How many ways are there for such movement? One dimensional is easy to solve as we can transform it to the two dimension dyke path which is an alternating explanation for the $Catalan$ $numbers$. However, is there way to solve the two dimensional cases? which means that there fours choices of movement which are either moving upward for one unit, downward for one unit, left for one unit or right for one unit, with a fixed number of steps, say $n$ and the last position of the lattice walk should have the farthest distance from the origin of the movement. The question is to find the number of such ways $a(n)$ in termes of $n $.