Compound patterns and generating functions: From basic waiting times to counts of occurrence

We derive an explicit, closed form expression for the double generating function of the corresponding counts of occurrence, within a finite time horizon, of the single patterns contained in a compound pattern. The expression is in terms of a basic single, and a basic joint, generating functions for which exact solutions exist in the literature. The single generating function is associated with the basic waiting time for the first occurrence of the compound pattern. The joint generating function is that for the waiting time to reach a given single pattern and the associated counts of occurrence, within that waiting time, of the single patterns contained in the compound pattern. The literature on patterns is huge. Also, there are results that establish links between generating functions for counts of occurrence of the single patterns contained in a compound pattern with generating functions of some more complex waiting times associated with that compound pattern. The latter waiting times are known in the literature with names such as sooner, or later waiting times, or generalisations of such. On the other hand, our result fills a gap in the literature by providing a neat link connecting the generating functions of the basic quantities associated with occurrence of compound patterns.