Parking Cars of Different Sizes

We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences. 1 The result. Parking functions were first introduced by Konheim and Weiss [5]. The original concept was that of a linear parking lot with n available spaces, and n cars with a stated parking preference. Each car would, in order, attempt to park in its preferred spot. If the car found its preferred spot occupied, it would move to the next available slot. A parking function is a sequence of parking preferences that would allow all n cars to park according to this rule. This definition is equivalent to the following formal definition: Definition 1.1. Let ~a = (a1, a2, . . . , an) be a sequence of positive integers, and let b1 ≤ b2 ≤ · · · ≤ bn be the increasing rearrangement of ~a. Then the sequence ~a is a parking function if and only if bi ≤ i for all indexes i. It is well known that the number of such parking functions is (n+ 1). This is Cayley’s formula for the number of labeled trees on n+1 nodes and Foata and Riordan found a bijective proof [3]. Stanley discovered the relationship between parking functions and non-crossing partitions [10]. Further connections have been found to other structures, such as priority queues [4], Gončarov polynomials [6] and hyperplane arrangements [11]. The notion of a parking function has been generalized in myriad ways; see the sequence of papers [2, 6, 7, 8, 12]. We present here a different generalization, returning to the original idea of parking cars. This time the cars have different sizes, and each takes up a number of adjacent parking spaces. Definition 1.2. Let there be n cars C1, . . . , Cn of sizes y1, . . . , yn, where y1, . . . , yn are positive integers. Assume there are ∑ n i=1 yi spaces in a row. Furthermore, let car Ci have the preferred spot ci. Now let the cars in the order C1 through Cn park according to the following rule: Starting at position ci, car Ci looks for the first empty spot j ≥ ci. If the spaces j through j + yi − 1 are empty, then car Ci parks in these spots. If any of the spots j + 1 through j + yi − 1 is already occupied, then there will be a collision, and the result is not a parking sequence.