A COMBINATORIAL INTERPRETATION OF THE AREA OF SCHRÖDER PATHS

An elevated Schr ̈oder path is a lattice path that uses the steps(1;1), (1;−1), and (2;0), that begins and ends on thex-axis, andthat remains strictly above thex-axis otherwise. The total areaof elevated Schr ̈oder paths of length 2n+ 2 satis es the recurrencefn+1=6fn−fn−1,n2, with the initial conditionsf0=1,f1=7.A combinatorial interpretation of this recurrence is given, by rst in-troducing sets of unrestricted paths whose cardinality also satis es therecurrence relation and then establishing a bijection between the setof these paths and the set of triangles constituting the total area ofelevated Schr ̈oder paths.