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

COMBINATORIAL THEORY OF Q,T-SCHRODER POLYNOMIALS, PARKING FUNCTIONS AND TREES Chunwei Song James Haglund 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)-Schroder Theorem is investigated. In the end, we derive a formula for the number of m-Schroder paths and study its q and (q, t)-analogues.