Enumeration of certain binary vectors

In 1986, W. S. Griffith [2] proposed reliability model and described a Markov chain approach for computing the reliability in the system. F. K. Hwang and S. Papastavridis gave a closed-form formula for this reliability in 1991, by counting the number of n-dimensional binary vectors containing exactly k non-overlapping m-tuples of consecutive ones. In 1988, T. M. Apostol [1] established recursive formulas and a generating function for a related problem, the enumeration of n-dimensional binary vectors containing exactly k isolated m-tuples of consecutive ones. In this paper we study two interrelated elementary enumerator functions in terms of which we are able to address the two enumeration problems mentioned above. This enables us to provide a unified approach to deriving closed-form formulas for both problems, and provides solutions to finer enumeration problems. We extend our results to the case of cyclic binary vectors, and briefly discuss the number of cyclic ordered partitions of a positive integer w. *This work was supported in part by NSA grant 904-93-H-3049, and by eelS Univ. of Nebraska Lincoln tThis work was supported in part by eCIS Univ. of Nebraska Lincoln Australasian Journal of Combinatorics 14 (1996), pp. 61-75