Relating Different Definitions of Linear Series on Tropical Curves

We compare strongly recursive tropical linear series as defined by Farkas, Jensen, and Payne with combinatorial limit linear series as defined by Amini and Gierczak. We show that strongly recursive tropical linear series of rank $r$ are combinatorial limit linear series of rank $r$ and construct a counterexample to the converse. We also present several extensions of our main result, including a simplification of the definition of combinatorial limit linear series and an investigation of their relationship with tropical linear series in the sense of Chang et al. Finally, we address the realizability of permutation arrays as local arrays of linear series on tropical curves. Finally, discuss the realizability of permutation arrays as local arrays of linear series on tropical curves.