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.
Highlights
Let C be a class of combinatorial structures
In (2), we studied decomposable combinatorial structures in the exp-log class with a given restricted pattern, where by a restricted pattern, we mean that the number of components of certain sizes are specified in advance
If the component generating function C(z) is of logarithmic type, we say that the corresponding structure F is in the exp-log class
Summary
Let C be a class of combinatorial structures. We call F a class of decomposable combinatorial structures over C if each element of F can be uniquely decomposed into a multiset of elements of C. In (2), we studied decomposable combinatorial structures in the exp-log class with a given restricted pattern, where by a restricted pattern, we mean that the number of components of certain sizes are specified in advance. We shall continue our work in (2) to study the probability of a random decomposable structure which has a given restricted pattern and a restricted size on its rth smallest component.
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