A Combinatorial Interpretation of the Area of Schröder Paths

An elevated Schröder path is a lattice path that uses the steps (1, 1), (1,−1), and (2, 0), that begins and ends on the x-axis, and that remains strictly above the x-axis otherwise. The total area of elevated Schröder paths of length 2n + 2 satisfies the recurrence fn+1 = 6fn − fn−1, n ≥ 2, with the initial conditions f0 = 1, f1 = 7. A combinatorial interpretation of this recurrence is given, by first introducing sets of unrestricted paths whose cardinality also satisfies the recurrence relation and then establishing a bijection between the set of these paths and the set of triangles constituting the total area of elevated Schröder paths.