Bijective Recurrences concerning Schro"der Paths

Consider lattice paths in Z2 with three step types: the up diagonal (1, 1), the down diagonal (1,−1), and the double horizontal (2, 0). For n ≥ 1, let Sn denote the set of such paths running from (0, 0) to (2n, 0) and remaining strictly above the x-axis except initially and terminally. It is well known that the cardinalities, rn = |Sn|, are the large Schröder numbers. We use lattice paths to interpret bijectively the recurrence (n+ 1)rn+1 = 3(2n− 1)rn− (n− 2)rn−1, for n ≥ 2, with r1 = 1 and r2 = 2. We then use the bijective scheme to prove a result of Kreweras that the sum of the areas of the regions lying under the paths of Sn and above the x-axis, denoted by ASn, satisfies ASn+1 = 6ASn − ASn−1, for n ≥ 2, with AS1 = 1, and AS2 = 7. Hence ASn = 1, 7, 41, 239, 1393, . . .. The bijective scheme yields analogous recurrences for elevated Catalan paths. Mathematical Reviews Subject Classification: 05A15 1 The paths and the recurrences We will consider lattice paths in Z whose permitted step types are the up diagonal (1, 1) denoted by U , the down diagonal (1,−1) denoted by D, and the double horizontal (2, 0) denoted by H. We will focus on paths that run from (0, 0) to (2n, 0), for n ≥ 1, and that never touch or pass below the x-axis except initially and terminally. Let Cn denote the set of such paths when only U-steps and D-steps are allowed, and let Sn denote the set of such paths when all three types are allowed. It is well known that the cardinalities cn = |Cn| and rn = |Sn|, for n ≥ 1, are the Catalan numbers and the large Schröder numbers, respectively. (See Section 4, particularly Notes 2 and 4.) Hence, here one might view the elements of Sn as elevated Schröder paths. Let ACn denote the sum of the areas of the regions lying under the paths of Cn and