Before revision:  The paired data structure in persistent homology as birth and death events of topological features add further complexity to conducting inference. To address these current problems, we propose to analyze the birth and death events using lattice paths .
After revision:  The paired data structure in persistent homology as birth and death in persistent diagrams, adds additional complexity to the development. In this paper, we present a new lattice path representation for persistent diagrams. A new exact statistical inference procedure is developed for lattice paths via combinatorial enumerations .