Enumeration of Generalized BCI Lambda-terms

We investigate the asymptotic number of elements of size n in a particular class of closed lambda-terms (so-called BCI(p)-terms) which are generalizations of lambdaterms related to axiom systems of combinatory logic. By deriving a differential equation for the generating function of the counting sequence we obtain a recurrence relation which can be solved asymptotically. We derive differential equations for the generating functions of the counting sequences of other more general classes of terms as well: the class of BCK(p)-terms and that of closed lambda-terms. Using elementary arguments we obtain upper and lower estimates for the number of closed lambda-terms of size n. Moreover, a recurrence relation is derived which allows an efficient computation of the counting sequence. BCK(p)-terms are discussed briefly. ∗Supported by ANR Magnum project (France) †Supported by ANR Boole project (France). This author’s work was partially carried out during her sabbatical leave at the Institute for Discrete Mathematics and Geometry, TU Wien, Austria. ‡Supported by FWF grant SFB F50-03 and ÖAD, grant F04/2012. the electronic journal of combinatorics 20(4) (2013), #P30 1