Connecting $k$-Naples parking functions and obstructed parking functions via involutions

Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained $k$-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set $PF_n$ of classical parking functions, independent of $k$. In this paper, we study $k$-Naples parking functions in the more general context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various parking function involutions to establish a bijection between the contained $k$-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of $k$-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.


Introduction
Parking functions are combinatorial objects introduced by Konheim and Weiss [8] in their study of hashing. For the purposes of this paper, a parking function can be defined as follows: Consider m cars c 1 , c 2 , . . . , c m trying to park, in that order, randomly on a one-way street with n parking spots, labeled 1, 2, . . . , n. For each j ∈ [m], car c j has a preferred parking spot a j ∈ [n] and will park there if the spot is unoccupied. If a j is occupied, then car c j tries to park at the next parking spot, and this process continues until c j reaches an unoccupied parking spot and parks there, or it terminates due to lack of available parking spots. This street can be modeled by a directed path I with vertices labeled 1, 2, . . . , n, and this sequence s ∈ [n] m of car preferences is called a classical (m, n)-parking function if all m cars manage to park through this process. Denote by P F (m, n) the set of classical (m, n)-parking functions. For m ≤ n, it has been shown that |P F (m, n)| = (n − m + 1)(n + 1) m−1 [2].
Classical parking functions have appeared in the study of topics such as tree enumeration [6], noncrossing partitions [9], and acyclic functions [10]. They have been generalized in various directions, including parking functions on digraphs [7] and Naples parking functions [1]. In this paper, we will explore a generalization of the latter called the k-Naples parking functions [3].
Fix integer 0 ≤ k ≤ n − 1. Consider the following modification to the parking rule for the aforementioned m cars, which will allow the cars to drive first backward (left), then forward (right) before parking. If car c j finds its preferred vertex a j occupied, then it travels to the nearest available vertex in the k spots before a j and parks there if such vertex exists. If all k vertices before a j are occupied, then c j drives to the nearest available vertex after a j and parks there if such vertex exists. The m-tuple (a 1 , a 2 , . . . , a m ) of car preferences is called a k-Naples (m, n)-parking function if all m cars are able to park via this process. Denote by P F (m, n; k) the set of k-Naples (m, n)-parking functions, with P F (m, n; 0) = P F (m, n).
There is an important subset B(m, n; k) ⊂ P F (m, n; k), called the contained k-Naples parking functions, consisting of those elements of P F (m, n; k) such that not all vertices before a j are occupied whenever a j ≤ k, for all j ∈ [m]. In other words, no car of B(m, n; k) has to "exit" the parking lot I in order to finish its backward search through k vertices for an available spot.
In the case m = n, denote P F n,k := P F (n, n; k) and B n,k := B(n, n; k), and |P F n,k | has the following recursive formula [3]: It was also proved in [3] that |B n,k | = (n + 1) n−1 independent of k, using a modified version of Pollack's technique [5]. Since this proof was not bijective, the authors posed the open problem of finding a bijection between B n,k and the set P F n := P F n,0 of classical (n, n)-parking functions that "preserves" in some manner certain parking function statistics such as ascents, descents, and ties. In addition, as the authors pointed out, it would be very difficult to derive a closed formula for |P F n,k | using the recursive formula (1), given that even special cases of (1) include quantities for which no closed formulae are known. Therefore, results on the size of |P F n,k | would be desirable.
Certain parking functions in which some vertices have been obstructed and hence unavailable for parking have been studied in [4], in the form of cars parking after a fixed trailer. We will tackle the previously raised questions through our effort to connect k-Naples parking functions with obstructed parking functions.
In this paper, we will study k-Naples parking functions using various involutions on these objects. For general m ≤ n, we introduce such an involution in Section 4.1 and apply it to construct a bijection between B(m, n; k) and P F (m, n) in Section 4.2. We then use this bijection to deduce that B(m, n; k) and P F (m, n) have the same number of ties, in Section 4.3. Lastly, in Section 5, we extend this bijection to an injection of P F (m, n; k) into LP F (m, n + k; k), the set of parking functions with m cars and n + k vertices whose first k vertices are obstructed. This will then allow us to obtain the upper bound with equality only if k = 0. Finally, in the case m = n, we combine (2) with a special case of the main result of [4] to obtain with equality only if k = 0.

Acknowledgments
The author would like to thank Pamela Harris for helpful conversations via email.

Preliminaries
For any two tuples u, v we denote by u ⊕ v the concatenation of u, v and we denote by len(v) the number of entries of v. Let m ≤ n ∈ N and let P F (m, n) denote the set of parking functions with m cars and n parking spots-called the classical (m, n)-parking functions. We regard the parking lot as a directed path I with vertices 1, 2, . . . , n, and we will denote each path on I as a tuple of its vertices. For vertices a < b ∈ [n] we denote by path(a, b) the path on I with a and b as left and right endpoints respectively. Now we give the definitions of the relevant parking functions introduced in [3].
Consider m cars trying to park on a directed path I with n vertices one after another, with car c j preferring vertex a j . If car c j finds a j unoccupied, it parks there. If a j is occupied, then c j drives backward from a j one vertex at a time to check if any vertices in the set A j,k : are available. If A j,k has unoccupied vertices, then c j parks in the available vertex a l ∈ A j,k closest to a j . If all vertices in A j,k are occupied, then c j drives forward and parks in the first unoccupied vertex after a j if it exists. If all cars are able to park under this rule, then we call the tuple f = (a 1 , a 2 , . . . , a m ) of car preferences a k-Naples (m, n)-parking function. Denote by P F (m, n; k) the set of all k-Naples (m, n)-parking functions.
The set of contained k-Naples (m, n)-parking functions B(m, n; k) is the subset of P F (m, n; k) such that if car c i has preference a i ≤ k, then some vertex in {1, 2, . . . , a i } has not been occupied by any car in {c 1 , c 2 , . . . , c i−1 }.
Definition 3.0.4. Let f = (a 1 , a 2 , . . . , a m ) ∈ P F (m, n; k). For all 1 ≤ j ≤ n − 1, we say that For the parking functions with obstructions we consider here, the parking rule is the same as that of the classical parking functions, but just with certain additional vertices unavailable for parking though they can still be preferred.
Definition 3.0.5. Fix integers 1 ≤ m ≤ n and 0 ≤ k ≤ n − 1. Consider m cars trying to park on a directed path I with n + k vertices one after another, where certain predetermined k consecutive vertices are occupied by obstructions and hence unavailable for parking. Each car c j will park at its preferred vertex a j if the spot is unoccupied. If a j is occupied either by an obstruction or by an earlier car, then c j will keep driving forward from a j until it finds an available vertex and parks there, or the process terminates due to lack of available vertices. The sequence f = (a 1 , a 2 , . . . , a m ) of car preferences is called a k-consecutive obstructed (m, n)parking function if all cars successfully park through this process. Denote by OP F (m, n+k; k) the set of k-consecutive obstructed (m, n)-parking functions.

Contained k-Naples parking functions
We will construct a bijection Ξ m,n;k between B(m, n; k) and P F (m, n) that "preserves" the distance traversed by each car before parking, taking into consideration the cars that park at or before their preferred spots and the cars that park after. Here, the cars of B(m, n; k) that park backward will correspond to the cars of P F (m, n) traversing no more than k vertices, while the cars of B(m, n; k) that park forward will correspond to the cars of P F (m, n) traversing more than k vertices. To deal with these two types of cars, it is useful to first construct an involution Φ m,n on P F (m, n) that, in some sense, "inverts" the configuration of the parked cars. Each f ∈ B(m, n; k) will be broken up into a sequence of sub-tuples, consisting of the preferences of backward parkers, the forward parkers, the backward parkers, and so on. Ξ m,n;k (f ) will be constructed one sub-tuple at a time, with Φ m,n being applied each time we move onto the next sub-tuple.

Involution on classical (m, n)-parking functions
We introduce an involution Φ m,n on P F (m, n) that "reflects" the parking configuration; call Φ m,n the parking reflection on P F (m, n). Let f ∈ P F (m, n). For each car c ∈ [m], let tpath(c) denote the path car c must traverse before parking; we call tpath(c) the traverse path of c.
For vertices i, j ∈ [n], we write i ∼ j if there exist cars c 1 , c 2 , . . . , c k such that tpath(c 1 ) contains i, tpath(c k ) contains j, and tpath(c l ) ∩ tpath(c l+1 ) = ∅. ∼ is a relation that yields "path components" that we will call parking components; every two vertices in a parking component are connected by a sequence of overlapping traverse paths. We will denote a parking component as a path, which is the union of all the relevant traverse paths.

Bijection between classical parking functions and contained k-Naples parking functions
Fix 0 ≤ k ≤ n− 1 and let B(m, n; k) denote the set of contained k-Naples parking functions with m cars. We will utilize the involutions Φ i,n for various i ∈ [m] to establish a bijection Ξ m,n;k between B(m, n; k) and P F (m, n); constructing an element of P F (m, n) from an element of B(m, n; k) will involve multiple "reflections" of parking configurations, depending on whether cars park backward or forward. Let f ∈ B(m, n; k). Note that cars in f never start by parking forward. We partition the m-tuple f into sub-tuples f = f 1 ⊕ f 2 ⊕ · · · ⊕ f d following their order in f so that: 1. f i consists of cars parking at or before their preferred spots for i odd.
2. f i consists of cars parking after their preferred spots for i even.
We will henceforth call f = f 1 ⊕ f 2 ⊕ · · · ⊕ f d the k-decomposition of f . The cars in Case 1 will correspond to the cars with traverse paths of length not exceeding k in Ξ m,n;k (f ), while the cars in Case 2 will correspond to the cars with traverse paths of length exceeding k in Ξ m,n;k (f ).
We construct Ξ m,n;k (f ) in stages by specifying the new preferences of the cars for each of f 1 , f 2 , . . . in that order; denote by Ξ m,n;k (f ) i the parking function obtained after f i . See Remark 4.2.1 for a rough picture of this procedure.
4. Continue this process as follows.

In general, the vertices occupied by cars in Ξ
Proof. It is easy to check that Ξ m,n;k (f ) ∈ P F (m, n) for all f ∈ B(m, n; k), where the occupied vertices in Ξ m,n;k (f ) are (a) the same as those of f if d is even We construct Ξ −1 m,n;k by reversing the procedure defining Ξ m,n;k . Let g ∈ P F (m, n). We produce a sequence Ξ −1 m,n;k (g) 1 , Ξ −1 m,n;k (g) 2 , . . ., Ξ −1 m,n;k (g) a of progressively longer tuples with Ξ −1 m,n;k (g) = Ξ −1 m,n;k (g) a .
Break up the m-tuple g into sub-tuples g =g 1 ⊕ g 1 satisfying one of the following: 1. All cars of g 1 have traverse path length not exceeding k, while the last car ofg 1 has traverse path length at least k + 1.
2. All cars of g 1 have traverse path length at least k + 1, while the last car ofg 1 has traverse path length not exceeding k.
Ξ −1 m,n;k (g) 1 will be a len(g 1 )-tuple defined in terms of g 1 . In Case 1, each car c 1 with preference p 1 in g 1 will have new preference n + 1 − p 1 in Ξ −1 m,n;k (g) 1 . In Case 2, each car c 1 with preference p 1 in g 1 will have new preference p 1 + k in Ξ −1 m,n;k (g) 1 .
Partition Φ len(g)−len(g1),n (g 1 ) into sub-tuples Φ len(g)−len(g1),n (g 1 ) =g 2 ⊕ g 2 satisfying one of the following: i. All cars of g 2 have traverse path length not exceeding k, while the last car ofg 2 has traverse path length at least k + 1.
ii. All cars of g 2 have traverse path length at least k + 1, while the last car ofg 2 has traverse path length not exceeding k.
For vertices i, j ∈ [n], we write i ∼ N j if there exist cars c 1 , c 2 , . . . , c a such that tpath N (c 1 ) contains i, tpath N (c a ) contains j, and tpath N (c l ) ∩ tpath N (c l+1 ) = ∅. The equivalence relation ∼ N yields "path components" that we shall call Naples components. We denote a Naples component as a path, which is the union of all the relevant Naples traverse paths.

Ξ m,n;k Preserves Ties in B(m, n; k)
Even though the correspondence Ξ m,n;k does not necessarily preserve the number of ties in a given f ∈ B(m, n; k), we can show that B(m, n; k) and Ξ m,n;k [B(m, n; k)] = P F (m, n) have the same number of ties, from which it follows that B n,k and P F n do as well.
For To record the change in ties due to Ξ m,n;k , we define the tie-change tuple as ∆ties(Ξ m,n;k , f ) = (e 1 , e 2 , . . . , e d−1 ), where m,n (f ). Any car c ′ with preference outside path(a, b) in f will have the same preference in Φ a,b m,n (f ). We will show that B(m, n; k) and P F (m, n) have the same number of ties, by defining an involution Ψ m,n;k on B(m, n; k) with the property that ∆ties(Ξ m,n;k , Ψ m,n;k (f )) = −∆ties(Ξ m,n;k , f ).
To aid us in this goal, we first define a simple involution ψ m,n;k on B(m, n; k) that only negates the last component e d−1 of ∆ties(Ξ m,n;k , f ).
In the degenerate case d = 1, we define ψ m,n;k (f ) = f , as f and Ξ m,n;k (f ) have the same ties. 1. If e d−1 = 1, that means Ξ m,n;k added a tie in the last component, so the preferences of the f d cars are decided by γ(b d−1 ) 1 , which "aims" certain cars in such a way that results in Ξ m,n;k subtracting a tie and which fixes the other car preferences.
2. If e d−1 = −1, that means Ξ m,n;k subtracted a tie in the last component, so the preferences of the f d cars are decided by γ(b d−1 ) 2 , which "aims" certain cars in such a way that results in Ξ m,n;k adding a tie and which fixes the other car preferences. . Furthermore, ψ m,n;k (f ) has the same Naples components as f , and ψ m,n;k (f ) has k-decomposition ψ m,n; have exactly the same ties. It remains to compare the last entries of ∆ties(Ξ m,n;k , f ) and ∆ties(Ξ m,n;k , ψ m,n;k (f )), to which end it suffices to check the cases e d−1 = ±1, for which d − 1 must necessarily be odd and hence the cars of f d must park forward. It is easy to see that the parking components of Ξ m,n;k (f ) | ′ d − 1 coincide with the Naples components of f | ′ d − 1.
Consider the Naples component L = (i, i + 1, i + 2, . . . , i + j) in f | ′ d − 1 containing f (b d−1 −1) = i+a for some 0 ≤ a ≤ j, with car b d−1 −1 eventually parking (backward) at vertex i. L is also a parking component in Ξ m,n;k (f ) | ′ d − 1 containing Ξ m,n;k (f )(b d−1 − 1) = i + j − a, with car b d−1 − 1 eventually parking at vertex i + j. Since e d−1 = ±1, there is a tie change occurring between cars b d−1 and b d−1 − 1, so L must be contained in tcomp(b d−1 ) and hence is not a tie in Ξ m,n;k (f ) and since the traverse path length of car , meaning that the preference ψ m,n;k (f )(c) makes the car c travel the same number k − a of steps left to enter L ′ in ψ m,n;k (f ) as it does right to enter L in Ξ m,n;k (f ), and hence c does park forward in ψ m,n;k (f ) because the same number k − a of occupied vertices precede i and follow i ′ + j. It also follows that tpath N (c) intersects the same Naples components in f and in ψ m,n;k (f ), up to permutation, so ψ m,n;k (f ) has the same Naples components as f . Since Ξ m,n;k [ψ m,n; so the preference ψ m,n;k (f )(c) makes the car c travel the same number k − a of steps left to enter L in f as it does right to enter L ′ in Ξ m,n;k [ψ m,n;k (f )], and hence ψ m,n;k (f ) has the same Naples components as f . Finally, Ξ m,n;k [ψ m,n; is indeed not a tie in Ξ m,n;k [ψ m,n;k (f )] as desired.
i=0 len(f d−i ),n;k to ψ j (f ) negates the last entry of ∆ties(Ξ m− j−1 i=0 len(f d−i ),n;k , ψ j (f )) while fixing other entries, and yields the same occupied vertices, Naples components, and kdecomposition as ψ j (f ), by Lemma 4.3.4. Hence Ψ m,n;k negates all the entries of ∆ties(Ξ m,n;k , f ) as desired, upon outputting Ψ m,n;k (f ) = Ψ m,n;k (f ) d via the above procedure.

Injection into the parking functions with obstructions
Let P F (m, n; k) denote the set of k-Naples parking functions with m cars on n vertices. We now extend Ξ m,n;k to an injection of P F (m, n; k) into LP F (m, n + k; k), the set of parking functions with m cars on n + k vertices where the first k parking spots are already occupied by obstructions. Again, let I denote the directed path with n + k vertices. Let OP F (m, n + k; k) ⊃ LP F (m, n + k; k) denote the set of parking functions with m cars on n + k vertices where certain k consecutive vertices are obstructed. We will first need to generalize the involution Φ i,n for i ≤ m to one on OP F (m, n + k; k).
Let f ∈ OP F (m, n+ k; k). We can define traverse paths of cars just as in P F (m, n), treating the obstructions as parked cars. Then we can define parking components in the same manner. Letk denote the path consisting of the k obstructed vertices. An obstruction component of f is a path on I that is either of the following: 1. a parking component of f that does not intersectk 2. the union ofk with all parking components of f that intersectk.
We now generalize Φ m,n to an involutionΦ m,n+k;k on OP F (m, n + k; k) as follows. Given f ∈ OP F (m, n + k; k), defineΦ m,n+k;k (f ) as follows. Let L = (i, i + 1, i + 2, . . . , i + j) be an obstruction component of f , with its reflection (n − i + 1, n − i, n − i − 1, . . . , n − i − j + 1) which we rewrite as the path (n − i − j + 1, n − i − j + 2, . . . , n − i + 1). For any car c preferring i + a in f for 0 ≤ a ≤ j, c will prefer n − i − j + a + 1 inΦ m,n+k;k (f ). In addition, if i + a, i + a + 1, . . ., i + a + k − 1 are the obstructed vertices in f for 0 ≤ a ≤ j, then the obstructed vertices in Φ m,n+k;k (f ) will be n − i − j + a + 1, n − i − j + a + 2, . . ., n − i − j + a + k. See Remark 5.0.1 for a rough picture ofΦ m,n+k;k . Thus defined, it is easy to see thatΦ m,n+k;k is an involution preserving the traverse path length of each car.
Remark 5.0.1. For any obstruction component L,Φ m,n+k;k maps L to its reflection L ′ = n + k + 1 − L, since I has length n + k. Furthermore, if L ′ = L + b, then any preference b i ∈ L is shifted to b i + b byΦ m,n+k;k .
Let f ∈ P F (m, n; k) with its k-decomposition f = f 1 ⊕ f 2 ⊕ · · · ⊕ f d . If f ∈ B(m, n; k), then defineΞ m,n;k (f ) = ι k [Ξ m,n;k (f )]. We now defineΞ m,n;k (f ) for the case f ∈ P F (m, n; k) − B(m, n; k); see Remark 5.0.3 for a rough picture of the procedure.
As in Section 4.2, we constructΞ m,n;k (f ) in stages by specifying the new preferences of the cars for each of f 1 , f 2 , . . . in that order; denote byΞ m,n;k (f ) i the parking function obtained after f i .
2. The obstruction components ofΞ m,n;k (f ) 2 coincide with the Naples components of f 1 ⊕f 2 , the difference being the lengthening by k obstructions due to ι k .
Proof. For g 1 ∈ B(m, n; k) and g 2 ∈ P F (m, n; k) − B(m, n; k),Ξ m,n;k (g 2 ) has cars preferring vertices in [k] whileΞ m,n;k (g 1 ) does not. HenceΞ m,n;k [B(m, n; k)] andΞ m,n;k [P F (m, n; k) − B(m, n; k)] do not intersect. We know that Ξ m,n;k is a bijection between B(m, n; k) and P F (m, n), and ι k naturally identifies P F (m, n) as the subset of LP F (m, n + k; k) where cars do not prefer the obstructed vertices. It remains to prove the claim for the caseΞ m,n;k | P F (m, n; k) − B(m, n; k). Let f ∈ P F (m, n; k) − B(m, n; k) with k-decomposition f = f 1 ⊕ f 2 ⊕ · · · ⊕ f d . Notice that the obstruction components ofΞ m,n;k (f ) 1 are the reflection of those of f | [len(f 1 )]. For Ξ m,n;k (f ) 2 = ι k [Φ len(f1),n (Ξ m,n;k (f ) 1 )] ⊕ f 2 , notice that Φ len(f1),n (Ξ m,n;k (f ) 1 ) has exactly the same obstruction components as f | [len(f 1 )] and that ι k merely "adds k obstructions to the left", so the cars of f 2 park inΞ m,n;k (f ) 2 exactly as they do in f .
In general, it is easy to observe the following: 1. For i even, the obstructions components ofΦ i−1 u=1 len(fu),n+k;k (Ξ m,n;k (f ) i−1 ) coincide with the Naples components of f restricted to [ If d is even, thenΞ m,n;k (f ) =Ξ m,n;k (f ) d has all its obstructions on the left. If d is odd, then Ξ m,n;k (f ) :=Φ m,n+k;k (Ξ m,n;k (f ) d ) ensures that all its obstructions are on the left. These two cases clearly do not intersect, since car m in one traverses more than k vertices while car m in the other does not. In both cases, the obstruction components ofΞ m,n;k (f ) coincide with the Naples components of f , excepting the k obstructions themselves. Since f is a parking function, Ξ m,n;k (f ) thus defined is an element of LP F (m, n + k; k).
To prove the injectivity ofΞ m,n;k , we first introduce some notations. Since car preferences are included sequentially in the procedure definingΞ m,n;k (f ), letΞ m,n;k (f ) c denote the stage in the construction ofΞ m,n;k (f ) after the preference of car c has been included.
u=1 len(f u )], thenΞ m,n;k (f ) c is a c-tuple that is a sub-tuple ofΞ m,n;k (f ) i . For any car a ≤ c ∈ [m], we denote by comp c (a, f ) the obstruction component ofΞ m,n;k (f ) c where a parks. We also denote by comp(a, f ) the obstruction component ofΞ m,n;k (f ) where car a parks. If p a , p b are the preferences of cars a, b respectively inΞ m,n;k (f ) c , let diff c (a, b; f ) = p b −p a which is their difference inΞ m,n;k (f ) c . If p ′ a , p ′ b are the preferences of cars a, b respectively in Ξ m,n;k (f ), let diff(a, b; f ) = p ′ b − p ′ a . We denote by tpath c (a,Ξ) the path car a must traverse before parking inΞ m,n;k (f ) c , and by tpath(a,Ξ) the traverse path of a inΞ m,n;k (f ).
Suppose that p a , p b are both in the same obstruction component inΞ m,n;k (f ) c , then we have diff c (a, b; f ) = diff d ′ (a, b; f ) for any d ′ > c, and diff c (a, b; f ) = diff(a, b; f ). In other words, the difference between any two preferences in the same obstruction component is preserved by the procedure definingΞ m,n;k (f ).
It is also clear that tpath c (a,Ξ), tpath d ′ (a,Ξ), and tpath(a,Ξ) all have the same length, for any d ′ > c. In other words, the procedure definingΞ m,n;k (f ) preserves the traverse path length of each car.
Assume for a contradiction that f = g ∈ P F (m, n; k)−B(m, n; k) butΞ m,n;k (f ) =Ξ m,n;k (g). Let f = f 1 ⊕ f 2 ⊕ . . . ⊕ f d1 and g = g 1 ⊕ g 2 ⊕ . . . ⊕ g d2 be their respective k-decompositions. Let j ∈ [m] be maximal such that car j has different preferences in f and g; label these preferences as p j and q j respectively, with f c1 containing p j and g c2 containing q j .
The above observations force d 1 = d 2 , c 1 = c 2 , and len(f l ) = len(g l ) for all l ∈ [d 1 ]. Let p ′ j and q ′ j denote the preferences of car j inΞ m,n;k (f ) c1 andΞ m,n;k (g) c1 respectively. By assumption, car l has the same preference in both f and g, for any l > j. We show that the obstruction components of car j will ultimately differ inΞ m,n;k (f ) andΞ m,n;k (g). Car j will have the same traverse path length t j ≥ 0 in bothΞ m,n;k (f ) c1 andΞ m,n;k (g) c1 . For every car c ∈ [j], c is in comp j (j, g) if and only if c is in comp j (j, f ), and we have diff j (j, c; g) = diff j (j, c; f ); otherwise, the cars in [j] would yield different obstruction components inΞ m,n;k (f ) | [j] andΞ m,n;k (g) | [j], which is a contradiction. Hence the car preference placements are the same in comp j (j, g) and comp j (j, f ), though these two components occupy different sets of vertices of I due to divergent preferences of car j.
Then the procedure will reflect comp j (j, f ) and comp j (j, g) a number of times; an even number of reflections will restore the original positions of comp j (j, f ) and comp j (j, g) provided no new cars will be added. If the procedure adds no new cars to comp j (j, f ) and comp j (j, g), then comp(j, g) and comp(j, f ) would still occupy different sets of vertices of I in Ξ m,n;k (g) andΞ m,n;k (f ) respectively. If the traverse path of car j ′ > j intersects comp j (j, f ) or comp j (j, g), then the cars in [j ′ ] would yield different obstruction components inΞ m,n;k (f ) | [j ′ ] andΞ m,n;k (g) | [j ′ ], because j ′ has the same preference in both situations by assumption.
Therefore, in all cases we haveΞ m,n;k (f ) =Ξ m,n;k (g), which is the desired contradiction. This shows thatΞ m,n;k is indeed injective. Proof. The bound |P F (m, n; k)| ≤ |LP F (m, n + k; k)| follows immediately from Theorem 5.0.6. For any k ≥ 1, no element of LP F (m, n + k; k) whose car 1 prefers an obstructed vertex will be inΞ m,n;k (P F (m, n; k)), so |P F (m, n; k)| < |LP F (m, n + k; k)|.
Lastly, consider the case m = n, where we denote LP F n+k,k := LP F (n, n + k; k) and P F n,k := P F (n, n; k). The main result (Theorem 1.2) of [4] yields the following special case Proposition 5.0.8. We have |LP F n+k,k | = (k + 1)(k + n + 1) n−1 .