Abstract
We consider two special subclasses of lambda-terms that are restricted by a bound on the number of abstractions between a variable and its binding lambda, the so-called De-Bruijn index, or by a bound on the nesting levels of abstractions, i.e., the number of De Bruijn levels, respectively.
 We show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to $Cn$ and $\tilde{C}n$, respectively, where the constants $C$ and $\tilde{C}$ depend on the bound that has been imposed. For the class of lambda-terms with bounded De Bruijn index we derive closed formulas for the constant. For the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of De Bruijn levels, we show quantitative and distributional results on the number of variables, as well as abstractions and applications, in the different De Bruijn levels and thereby exhibit a so-called "unary profile" that attains a very interesting shape. 
 Our results give a combinatorial explanation of an earlier discovered strange phenomenon exhibited by the counting sequence of this particular class of lambda-terms.
Highlights
1.1 Motivation Lambda-terms are objects stemming from lambda-calculus and can be seen as combinatorial objects with a simple description
In the lambda-directed acyclic graphs (DAGs), the De Bruijn indices and levels are visible: The De Bruijn index of a variable v is the number of unary nodes we find on the path from v to its binding lambda in the skeleton of the lambda-DAG, where the last unary node on the path has to be counted as well
In the last section we investigate the distribution of the different types of nodes in lambda-DAGs with bounded number of De Bruijn levels among the separate levels throughout the DAG
Summary
1.1 Motivation Lambda-terms are objects stemming from lambda-calculus and can be seen as combinatorial objects with a simple description. The electronic journal of combinatorics 26(4) (2019), #P4.47 is not well understood. They can be seen as words (sequence of symbols) or graphs and the combinatorially most natural way to define an enumeration problem is to ask for the number of terms with a given number of symbols or vertices, respectively. This problem appears to be very intriguing, as standard techniques fail. Our motivation to perform the present investigation is to shed light on this oddity and to give a combinatorial explanation for this phenomenon
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