Schröder Paths, Their Generalizations and Knot Invariants

We study some kinds of generalizations of Schröder paths below a line with rational slope and derive the $q$-difference equations that are satisfied by their generating functions. As a result, we establish a relation between the generating function of generalized Schröder paths with backwards and the wave function corresponding to colored HOMFLY-PT polynomials of torus knot $T_{1,f}$. We also give a combinatorial proof of a recent result by Stošić and Sułkowski, in which the standard generalized Schröder paths are related to the superpolynomial of reduced colored HOMFLY-PT homology of $T_{1,f}$.