CYCLE SYSTEMS

Abstract

In this paper we show that the composition (symmetric difference) of cycles is well-defined. So, such a collection {…, Ci, Ck,.…} of cycles with a composition operator, °, is a matroid. As such, it has sets of independent, or basis, cycles that determine its rank r. This paper is concerned with independent and dependent sets of cycles within a cycle system. In particular, we enumerate the number of all possible basis sets in any cycle system of rank r ≤ 6. Then we use a generating function to establish that the ratio of basis sets to all possible r element sets approaches c, 0.287 < c < 0.289.