Vacillating parking functions

For any integers 1≤ k≤ n, we introduce a new family of parking functions called k-vacillating parking functions of length n. The parking rule for k-vacillating parking functions allows a car with preference p to park in the first available spot in encounters among the parking spots numbered p, p-k, and p+k (in that order and if those spots exists). In this way, k-vacillating parking functions are a modification of Naples parking functions, which allow for backwards movement of a car, and of ℓ-interval parking functions, which allow a car to park in its preference or up to ℓ spots in front of its preference. Among our results, we establish a combinatorial interpretation for the numerator of the nth convergent of the continued fraction of √(2), as the number of non-decreasing 1-vacillating parking functions of length n. Our main result gives a product formula for the enumeration of k-vacillating parking functions of length n based on the number of 1-vacillating parking functions of smaller length. We also give closed formulas for the number of k-vacillating parking functions of length n which are monotonically increasing or decreasing. We conclude with some directions for further research.