@article{Xin2025, 
author = {Hua Xin},
title = {Lattice points of flow polytopes related to caracol graphs},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {10},
pages = {6141-6175},
keywords = {flow polytopes, caracol graphs, lattice points, lattice paths, Kostant partition functions},
url = {https://www.sciopen.com/article/10.3934/era.2025272},
doi = {10.3934/era.2025272},
abstract = {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.}
}