Geodesic
The Size of the rth Smallest Component in Decomposable Structures with a Restricted Pattern
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) (2007).

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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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     author = {Dong, Li and Gao, Zhicheng and Panario, Daniel},
     title = {The {Size} of the rth {Smallest} {Component} in {Decomposable} {Structures} with a {Restricted} {Pattern}},
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Dong, Li; Gao, Zhicheng; Panario, Daniel. The Size of the rth Smallest Component in Decomposable Structures with a Restricted Pattern. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) (2007). doi : 10.46298/dmtcs.3526. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3526/

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