Parking Cascades: From the Simplest Sequence to Motzkin and Catalan

We introduce $k$-cascading parking functions, a parametrized variant of parking functions in which cars form bumping cascades of up to $k \geq 0$ cars. Setting $k = 0$ recovers classical parking functions, whereas $k = 1$ recovers MVP parking functions. Although parking functions and cascading parking functions are equivalent as sets, they are generally distinct as maps. Therefore, in this paper we consider the enumeration of the fibers of their outcomes. Our main result is a recursive, permutation pattern-based formula for the size of the fiber of any given permutation, for any given $k \geq 0$. When specialized to the longest word, the formula reduces to a family of integer sequences that interpolate between the simplest sequence ($k=0$), the Motzkin numbers ($k = 1$), and the Catalan numbers ($k\geq n-1$). When specialized to the set of layered permutations, the formula gives new combinatorial interpretations for the row sums of certain convolution triangles, including Motzkin and Catalan convolution triangles.