| A decomposable combinatorial structure consists of simpler objects called components which by themselves can not be further decomposed. For example, permutations decompose into cycles, graphs into connected components, and polynomials over finite fields into irreducible factors. The size of an object is the number of its elements. For example, a permutation on n objects, a graph on n vertices and an n-degree polynomial over a finite field have all size n. In this thesis, we study the decomposable combinatorial structures with given restricted patterns. A restricted pattern for a decomposable object of size n fixes the number of components whose sizes are specified in the restricted pattern. In other words, partial information of this object is known. For instance, the number of permutations with a certain factorization pattern is used in the study of the cycle index of the symmetric group.;We study two important classes of decomposable combinatorial structures: exp-log class and alg-log class. For the structures in these two classes, using the method of analysis of singularities, we derived an asymptotic expression for the probability that a decomposable structure of size n has a given restricted pattern. In addition to restricted pattern, we estimate the probability that the size of the random decomposable combinatorial structure's rth smallest component is bigger than k, for r, k given integers. Furthermore, we estimate the first moment of the size of the rth smallest component in both exp-log class and alg-log class with restricted patterns. In the exp-log class, we also obtain a result for higher moments.;Since decomposable combinatorial structures are traditionally divided into labeled and unlabeled cases, we provide results for the two cases throughout this thesis. Applying a combinatorial method, we obtain the generating function of combinatorial structures with a given restricted pattern for both labeled and unlabeled objects. Furthermore, in addition to the restricted pattern, we derive the generating function when we restrict the size of the rth smallest component of decomposable combinatorial structure. |
