  Persistent homology has undergone significant development in recent years.
However, one outstanding challenge is to build a coherent statistical inference
procedure on persistent diagrams. In this paper, we first present a new lattice
path representation for persistent diagrams. We then develop a new exact
statistical inference procedure for lattice paths via combinatorial
enumerations. The lattice path method is applied to the topological
characterization of the protein structures of the COVID-19 virus. We
demonstrate that there are topological changes during the conformational change
of spike proteins.
