Combinatorial theory of q,t-Schröder polynomials, parking functions and trees

We study various aspects of lattice path combinatorics. A new object, which has Dyck paths as its subset and is named Permutation paths, is considered and relative theories are developed. We prove a class of tree enumeration theorems and connect them to parking functions. The limit case of ( q,t)-Schröder Theorem is investigated. In the end, we derive a formula for the number of m-Schröder paths and study its q and (q,t)-analogues