Distribution of Patterns of Constrained Length in Binary Sequences

Abstract

On a finite sequence of binary (0-1) trials we define a random variable enumerating patterns of length subject to certain constraints. For sequences of independent and identically distributed binary trials exact probability mass functions are established in closed forms by means of combinatorial analysis. An explicit expression of the mean value of this random variable is obtained. The results associated with the probability mass functions are extended on sequences of exchangeable binary trials. An application in Information theory concerning counting of a class of run-length-limited binary sequences is provided as a direct byproduct of our study. Illustrative numerical examples exemplify further the results.