On the Number of Lambda Terms With Prescribed Size of Their De Bruijn Representation

John Tromp introduced the so-called ’binary lambda calculus’ as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with m free indices and of size n (encoded as binary words of length n) is o ( n−3/2τ−n ) for τ ≈ 1.963448 . . .. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with m free indices, the number of terms of size n is Θ ( n−3/2ρ−n ) with some class dependent constant ρ, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given.