Counting Runs of Ones and Ones in Runs of Ones in Binary Strings

摘要

Consider a binary string (a symmetric Bernoulli sequence) of length . For a positive integer , we exactly enumerate, in all? possible binary strings of length , the number of all runs of 1s of length (equal, at least)? and the number of 1s in all runs of 1s of length at least . To solve these counting problems, we use probability theory and we obtain simple and easy to compute explicit formulae as well as recursive schemes, for these potential useful in engineering numbers.