Generating Function Methods for Run and Scan Statistics

Runs and pattern statistics have found successful applications in various fields. Many classical results of distributions of runs were obtained by combinatorial methods. As the patterns under study become complicated, the combinatorial complexity involved may become challenging, especially when dealing with multistate or multiset systems. Several unified methods have been devised to overcome the combinatorial difficulties. One of them is the finite Markov chain imbedding approach. Here we use a systematic approach that is inspired by methods in statistical physics. In this approach the study of run and pattern distributions is decoupled into two easy independent steps. In the first step, elements of each object (usually represented by its generating function) are considered in isolation without regards of elements of the other objects. In the second step, formulas in matrix or explicit forms combine the results from the first step into a whole multi-object system with potential nearest neighbor interactions. By considering only one kind of object each time in the first step, the complexity arising from the simultaneous interactions of elements from multiple objects is avoided. In essence the method builds up a higher level generating function for the whole system by using the lower level of generating functions from individual objects. By dealing with generating functions in each step, the method usually obtains results that are more general than those obtained by other methods. Examples of different complexities and flavors for run- and pattern-related distributions will be used to illustrate the method.