Bijective Recurrences concerning Schrr Oder Paths

Consider lattice paths in Z 2 with three step types: the up diagonal (1; 1), the down diagonal (1; ?1), and the double horizontal (2; 0). For n 1, let S n 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, r n = jS n j, are the large Schrr oder numbers. We use lattice paths to interpret bijectively the recurrence (n + 1)r n+1 = 3(2n ? 1)r n ? (n ? 2)r n?1 , for n 2, with r 1 = 1 and r 2 = 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 S n and above the x-axis, denoted by AS n , satisses AS n+1 = 6AS n ? AS n?1 ; for n 2, with AS 1 = 1, and AS 2 = 7. 1 The paths and the recurrences We will consider lattice paths in Z 2 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 C n denote the set of such paths when only U-steps and D-steps are allowed, and let S n denote the set of such paths when all three types are allowed. It is well known that the cardinalities c n = jC n j and r n = jS n j, for n 1, are the Catalan numbers and the large Schrr oder numbers, respectively. (See Section 4, particularly Notes 2 and 4.) Hence, here one might view the elements of S n as elevated Schrr oder paths. Let AC n denote the sum of the areas of the regions lying under the paths of C n and 1