Within population biology, models are often analyzed for the net reproduction number or other generalized target reproduction numbers, which describe the growth or decline of the population based on specific mechanisms. This is useful in determining the strength and efficiency of control measures for inhibiting or enhancing population growth. The literature contains many algebraic and combinatorial approaches for deriving the net reproduction number and generalized target reproduction numbers from digraphs and associated matrices. Finding, categorizing, and counting the permutations of disjoint cycles, or cycles unions is a requirement of the Cycle Union approach by Lewis et al. (2019). These cycles and subsequent cycle unions can be found via the digraphs and associated matrices. We developed cycle counting patterns for targeting fertilities within Leslie Matrices, Lefkovitch Matrices, Sub-Diagonal Lower Triangle Transition Matrices, and Lower Triangle Transition Matrices to serve as a foundation for future work. Presented are the counting patterns and closed-form summations of the cycle unions.