Counting General Dyck Paths With Respect To Some Parameters

Combinatorial mathematics,also known as combinatorial theory,combinatorics,is an important branch of mathematics.Counting problem is a problem which has been studied the most in combination and appears in all branches of mathematics.Many scholars have made a large number of research on the lattice counting.Lattice is taken into account in the plane rectangular coordinate system usually.Limiting the step set can obtain different types of lattice.A Generalized Dyck Path is a path in the first quadrant which begins at the origin,ends at(2n,0),and consists of steps(k,k),called rises,and(1,-1),called falls.This paper consider the problem of counting steps and peaks,return steps,height of first peak,valleys,peaks before and after the first return step and calculating the expectation and variance of low peak,return step and low valley on Generalized Dyck Path.We mainly use symbolic method to get generating function.Moreover,we use Lagrange inversion theorem and theorems of log-convex sequence to get the corre-sponding counting results.