Enumeration Of K-colored Skew Motzkin Path And Equivalence Classes Of Shuffle Of Motzkin Paths

Lattice path is an important research direction of combinatorial mathematics,which is closely related to statistics,molecular biology and other emerging disciplines.Motzkin path is one of the important subjects of lattice path and rich research results have been obtained.This paper is divided into two parts.In the first part,we introduce the conception of k-colored skew Motzkin paths.By using the Language inversion theorem and symbolic method,we give the generating function and counting formula,according to the number of length,up steps,left steps,peaks,double rise and so on.We also give the explaination between the k-colored skew Motzkin paths and k-colored skew Dyck paths,skew Motzkin paths,skew Dyck paths and so on.In the second part,we investigate the equivalence classes on shuffle of Motzkin paths.We study equivalence classese whose length at most 2 such as h_?-equivalence classes,?-equivalence classes,?b-equivalence classes,??-equivalence classes,?h_?-equivalence classesand give these generating functions.