We introduce the class of bilateral parking procedures on the integer line. While cars try to park in the nearest available spot to their right in the classical case, we consider more general parking rules that allow cars to use the nearest available spot to their left. We show that for a natural subclass of local procedures, the number of corresponding parking functions of length $r$ is always equal to $(r+1)^{r-1}$. The setting can be extended to probabilistic procedures, in which the decision to park left or right is random. We finally describe how bilateral procedures can naturally be encoded by certain labeled binary forests, whose combinatorics shed light on several results from the literature.
Classical parking functions are a central object in enumerative and algebraic combinatorics. They are connected to various structures such as noncrossing partitions, hyperplane arrangements, and many others: see for instance the survey [17] and references therein. The corresponding parking procedure P right on Z was originally defined as an elementary hashing procedure, cf. [10].
We recall its definition: r cars want to park on an empty one-way street with spots labeled by 1, 2, . . . , r from left to right. The cars arrive successively, and the ith car has a preferred spot a i . If this spot is available, it parks there, and if not it parks in the nearest available spot to the right. The sequence (a 1 , . . . , a r ) is called a parking function if at the end, all cars managed to park. The number of parking functions for r cars is given by the simple formula (r + 1) r-1 [10].
The following characterization is well known: (a 1 , . . . , a r ) is a parking function if and only if for any k = 1, . . . , r, there are at least k indices i such that 1 ≤ a i ≤ k. Parking function generalizations have often relied on this description. This is the case of G-parking functions and u-parking functions, see [17]. Recently, several variations were also considered by varying different aspects of the procedure, see for instance [3] and references therein.
In this work, we will consider the extensions of the parking procedure obtained by simply changing the “nearest available spot to the right” condition: we will also allow one to park to the nearest available spot to the left. This explains the name bilateral for our class of procedures.
Let us give already two examples of possible rules. We need only describe where to park when one’s preferred spot is occupied:
• P closest : If the nearest available spot to the right is (weakly) closer to a i than the nearest available spot to the left, park there; otherwise park to the left.
• P prime : If the total number of cars parked between the nearest available spots to the left and to the right is a prime number, park to the right; otherwise park to the left. In general we will describe parking procedures P as functions associating a finite subset of Z of cardinality r to a preference word a 1 • • • a r , so that P(a 1 • • • a r ) is the set of occupied spots after r cars have parked. If it is equal to {1, . . . , r} then we will say that a 1 • • • a r is a P-parking function. The notion of bilateral procedure is then easy to encode as conditions on the function P.
We will then state two conditions to determine the subclass of local procedures: roughly put, these say that left/right decisions must be invariant under translation, and depend only on the cars that parked around the desired spot. We then obtain the following striking enumerative result.
Theorem 1.1. Let P be a bilateral, local parking procedure. Then the number of P-parking functions of length r is given by (r + 1) r-1 .
In particular this holds for the classical procedure denoted P right , for P closest and P prime defined above (and for uncountably many other procedures). One may interpret Theorem 1.1 as a kind of discrete universality result: a unique formula holds for a large classe of procedures where only local conditions are imposed. The proof of this result will follow from Pollak’s argument in the classical case, based on a cyclic procedure derived from the original one.
We can naturally add randomness to a bilateral procedure: instead of choosing either left or right, pick probabilities for the two options. Each preference word now has a certain probability to be P-parking. For a particular procedure, we will see that these probabilities, suitably normalized, coincide with the family of remixed Eulerian numbers studied by Vasu Tewari and the author [12]. The notion of local procedure extends to the probabilistic setting, and a version of Theorem 1.1 holds in this case.
From the point of view of bijective combinatorics, we show that bilateral procedures are naturally encoded by certain pairs of labeled forests. This bijection naturally lifts the original parking procedure, and is directly connected to the outcome function of the procedure, which encodes the order in which the cars parked. This encoding recovers some known results, including in the classical case.
Outline. We first describe rigorously the parking procedures P that we consider in Section 2. We will focus on bilateral procedures, and then introduce the subclass of local procedures in Section 3. We will then see that the enumeration (r + 1) r-1 is in a sense universal for local procedures, see Theorem 1.1. We then explain how to define a probabilistic version of our procedures in Section 4. We describe some natural connection with the combinatorics of binary trees via a natural encoding in Section 5. We finally briefly define a “colored” version of the model in the last section.
As stated in the introduction, the list of parking spo
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