Lattice points of flow polytopes related to caracol graphs

Flow polytopes are fundamental objects in algebraic combinatorics. In this paper, we study the enumeration of lattice points in flow polytopes associated with $(a_1,a_2)$-caracol graphs on $n+2$ vertices. Our main result establishes a closed-form expression for the number of lattice points by constructing an explicit combinatorial bijection between Dyck paths and the integer lattice points of the two-parameter family of polytopes $(a_1,a_2)$-Car ${}_{n+1}$, using pseudo-ladder diagrams together with vector partition techniques. When $a_2=1$, the lattice point sequence of caracol polytopes coincides with the OEIS sequence A126216 (The On-Line Encyclopedia of Integer Sequences), which enumerates Schröder paths of semilength $n$ with exactly $k$ peaks. Furthermore, we establish a bijection between Schröder paths and the integer lattice points of the two-parameter family of polytopes $(a_1,a_2)$-Car ${}_{n+1}$. All bijections are implemented as explicit algorithms in Python, with the complete source code provided in the appendix to ensure reproducibility.