### The Size of the rth Smallest Component in Decomposable Structures with a Restricted Pattern

*Li Dong, Zhicheng Gao, Daniel Panario*

#### Abstract

In our previous work [paper1], we derived an asymptotic expression for the probability that a random decomposable combinatorial structure of size n in the &exp;-log class has a given restricted pattern. In this paper, under similar conditions, we provide the probability that a random decomposable combinatorial structure has a given restricted pattern and the size of its rth smallest component is bigger than k, for r,k given integers. Our studies apply to labeled and unlabeled structures. We also give several concrete examples.

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