The volume of the caracol polytope

We give a combinatorial interpretation of the Lidskii formula for flow polytopes and use it to compute volumes via the enumeration of new families of combinatorial objects which are generalizations of parking functions. Our model applies to recover formulas of Pitman and Stanley, and compute volumes of previously seemingly unapproachable flow polytopes. A highlight of our model is that it leads to a combinatorial proof of an elegant volume formula for a new flow polytope which we call the caracol polytope. We prove that the volume of this polytope is the product of a Catalan number and the number of parking functions.