A Survey of Parking Functions
A parking function of length n is a sequence a1, a2, . . . , an of positive integers whose increasing rearrangement b1 ≤ b2 ≤ · · · ≤ bn satisfies bi ≤ i. Parking functions go back to Pyke in 1959; the term “parking function” and the connection with the parking of cars is due to Konheim and Weiss (1966).Pollak gave an elegant proof that the number of parking functions of length n is (n + 1)n−1. There are close connections between parking functions and other combinatorial objects such as trees, non-crossing partitions, and the Shi hyperplane arrangement. Parking functions arise in several unexpected algebraic areas, such as representations of the symmetric group and Haiman’s theory of diagonal harmonics. Parking functions also have a number of natural generalizations which fit together in a nice way. We will survey these aspects of the theory of parking functions.