$λ$ -项的组合学：一种自然方法
Combinatorics of $λ$-terms: a natural approach
摘要：
我们认为组合方面的$λ$ -条款的模型的基础上德布鲁因指数，其中每个建筑构造函数的大小为1。令人惊讶的是，$λ$项的计数序列也对应于两个二叉树家族，即黑白树和无标记树。我们提供了一个建设性的证明，这一事实表现出适当的双射。此外，我们确定的序列的Motzkin数与计数序列的中性$λ$ -条款，给出了一个双射，因此，结果在一个精确的大小采样器为后者的基础上精确的大小采样器为Motzkin树的Bodini等alli。利用强大的分析组合学理论，我们陈述了几个结果关于$λ$ -项的渐近增长率在中性，正常，和头部范式。最后，我们调查的渐近密度的$λ$ -条款包含任意固定的子项显示，除其他外，强规范化或typeable条款是渐近忽略不计的所有$λ$ -条款。
英文摘要：
We consider combinatorial aspects of $λ$-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for $λ$-terms corresponds also to two families of binary trees, namely black-white trees and zigzag-free ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we identify the sequence of Motzkin numbers with the counting sequence for neutral $λ$-terms, giving a bijection which, in consequence, results in an exact-size sampler for the latter based on the exact-size sampler for Motzkin trees of Bodini et alli. Using the powerful theory of analytic combinatorics, we state several results concerning the asymptotic growth rate of $λ$-terms in neutral, normal, and head normal forms. Finally, we investigate the asymptotic density of $λ$-terms containing arbitrary fixed subterms showing that, inter alia, strongly normalising or typeable terms are asymptotically negligible in the set of all $λ$-terms.
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