A Classic Proof of a Recurrence for a Very Classical Sequence

Richard Stanley St96] has recently narrated the fascinating story of how the classical Schrr oder Sch1870] numbers s(n) are even more classical than has been believed before. They (at least s(10) = 103049) have been known to Hipparchus (190-127 B.C.). Stanley recalled the three-term linear recurrence ((Co64]; Co74], p. 57) (1) 3(2n ? 1)s(n) = (n + 1)s(n + 1) + (n ? 2)s(n ? 1) (n 2); s(1) = s(2) = 1; and stated that \no direct combinatorial proof of this formula seems to be known." The purpose of this note is to ll this gap. The present proof reeects the ideas of our great master, Marcel-Paul Sch utzenberger (1920-1996), who taught us that every algebraic relation is to be given a combinatorial counterpart and vice versa. The methodology has been vigorously and successfully pursued by the Ecole bordelaise (e.g., Cor75], Vi85].) The recurrence (1) is tantalizingly similar to the linear recurrence (2) 2(2n ? 1)c(n) = (n + 1)c(n + 1) (n 1); c(1) = 1; that is obviously satissed by the Catalan numbers c(n) = (1=n) ? 2n?2 n?1. Our proof is inspired by R emy's elegant combinatorial proof Re85] of (2) shown to us by Viennot Vi82]. Recall ((Co74], pp. 56-57) that a Schrr oder tree T is either the tree consisting of its root alone T = r, or an ordered tuple r; T 1 ; : : : ; T l ], where l 2 and T 1 , : : : , T l are smaller Schrr oder trees. The rst symbol r is called the root of T and the roots of T 1 , : : : , T l are called the sons of T. A son-less node is called a leaf. The number of Schrr oder trees with n leaves is denoted by s(n). The rst values of the s(n)'s appear in Table 1.