Generalized Small Schröder Numbers

We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from (0, 0) to (2n, 0) with the step set of {(k, k), (l,−l), (2r, 0) | k, l, r ∈ P}, where P is the set of positive integers, which never goes below the x-axis, and with no horizontal steps at level 0. We find a bijection between 5-colored Dyck paths and generalized small Schröder paths, proving that the number of generalized small Schröder paths is equal to ∑n k=1N(n, k)5 n−k for n > 1.