Binary vectors with exactly k nonoverlapping m-tuples of consecutive ones

Recently, Apostol studied the number of binary vectors in n-space containing exactly k isolated m-tuples of consecutive ones. He derived recursive formulas and generating functions for determining this number. In this note the authors study the same problem except changing the word ‘isolated’ to ’nonoverlapping’, meaning that two m-tuples are allowed to be adjacent. They give a closed-form solution for the concerned number. The authors also extend the present result to cyclic binary vectors and apply it to a reliability model.