Quantitative aspects of linear and affine closed lambda terms

Affine λ-terms are λ-terms in which each bound variable occurs at most once and linear λ-terms are λ-terms in which each bound variable occurs once. and only once. In this paper we count the number of closed affine λ-terms of size n, closed linear λ-terms of size n, affine β-normal forms of size n and linear β-normal forms of size n, for different ways of measuring the size of λ-terms. From these formulas, we show how we can derive programs for generating all the terms of size n for each class. The foundation of all of this is specific data structures, which are contexts in which one counts all the holes at each level of abstractions by λ’s.