A Generalization of Parking Functions Allowing Backward Movement

Alex Christensen; Andr\'es Ramos Rodr\'iguez; Gordon Rojas Kirby; Joseph Rennie; Marissa Loving; Pamela E. Harris; Zakiya Jones

REVIEW 3 major objections 5 minor 9 references

A Generalization of Parking Functions Allowing Backward Movement

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves a recursive formula that counts $k$-Naples parking functions, in which a car whose preferred spot is taken may first back up to $k$ spaces before driving on.

desk verdict A useful new family of parking functions with a plausible recursion, but two proofs need work before the results are fully trustworthy. read the letter →

arxiv 1908.07658 v1 pith:J2DSOABG submitted 2019-08-21 math.CO

Alex Christensen , Pamela E. Harris , Zakiya Jones , Marissa Loving , Andrés Ramos Rodríguez , Joseph Rennie , Gordon Rojas Kirby This is my paper · ORCID

classification math.CO MSC 05A1505A19

keywords parkingfunctionsk-NaplesbackwardmovementrulecontainedsignatureDyckpathsk-latticerecursiveenumerationcircleargument

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces $k$-Naples parking functions: parking preferences for $n$ cars in which a car that finds its preferred spot occupied may first check up to $k$ spaces to the west before driving east. Its central result is a recursion that determines the number of these preferences for every length $n$ and every $0\le k\le n-1$, interpolating between the classical parking-function count $(n+1)^{n-1}$ at $k=0$ and the full set $n^n$ when $k\ge n-1$. The paper also proves that a preference is a $1$-Naples parking function exactly when a simple transformation of it is a classical parking function, and it bijects decreasing $k$-Naples parking functions with $k$-lattice paths, a family of signature Dyck paths. A sympathetic reader would care because the recursion turns a new generalization into a sequence that can be computed exactly and connected to known combinatorial families.

What carries the argument

The load-bearing mechanism is a two-sided decomposition of the parking lot at the place where the last car parks. The cars ending up west of that place form a $k$-Naples parking function of length $i$; the cars ending up east form a contained $k$-Naples parking function, defined as one that would leave an extra spot $0$ west of spot $1$ empty. The count of contained functions is $(n+1)^{n-1}$ for every $k$, obtained by arranging $n+1$ spots on a circle, parking all $n$ cars, and rotating preferences so that spot $0$ is the one left empty; this circle step is what converts the east block into the factor $(n-i+1)^{n-i-1}$. For the $k=1$ characterization the central object is the map $T$ that subtracts one from a preferred spot the moment that spot has already been claimed earlier in the tuple, and for the lattice-path result it is the $k$-lattice path, a Dyck path whose height may exceed the diagonal by up to $k$.

What would settle it

Brute-force all $7^7$ length-7 preferences under the $1$-Naples rule, record the final parking spot of the seventh car, and compare the number landing in each spot $i+1$ with the summand $\binom{6}{i}\min(i+2,7)|PF_{i,1}|(7-i)^{5-i}$ from Theorem 1.1; a mismatch for any $i$ would show the independent-left-block assumption fails.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for $0\le k\le n-1$, the number of $k$-Naples parking functions of length $n+1$ is $$|PF_{n+1,k}|=\sum_{i=0}^n \binom{n}{i}\min(i+1+k,n+1)|PF_{i,k}|(n-i+1)^{n-i-1}.$$ The proof fixes the spot $i+1$ where the last car parks, splits the earlier $n$ cars into those that end up west and east of that spot, counts the west block as a $k$-Naples parking function of length $i$, the east block as a contained $k$-Naples parking function (one that would leave an extra west spot empty), and counts the last car's allowed preferences as $\min(i+1+k,n+1)$. For $k=1$ the paper proves that a preference $\alpha$ is a Naples parking function exactly when the map $T$ that lowers every repeated preferred entry by $1$ sends $\alpha$ to a classical parking function, and for decreasing preferences it gives a bijection to $k$-lattice paths, i.e. Dyck paths allowed to stay up to $k$ units above the diagonal.

Load-bearing premise

The recursion's count of the west block assumes that the cars that end up in the first $i$ spots have preferences forming a $k$-Naples parking function of length $i$ on their own, with no car preferring a spot east of the cut and then backing into that block.

Editorial extensions

If this is right

For every pair $(n,k)$ with $0\le k\le n-1$, the recursion determines $|PF_{n,k}|$ by a finite computation, so no exhaustive test of $n^n$ preferences is needed.
At $k=0$ the formula reduces to the classical recursion for parking functions, and at $k=n-1$ it yields $n^n$, so the two endpoints of the interpolation are consistent.
The closed forms $|PF_{n,n-1}|=n^n$ and $|PF_{n,n-2}|=n^n-n^{n-2}$ pin down the top two diagonals of the enumeration table, leaving the intermediate cases to the recursion.
The transformation $T$ gives a membership test for $1$-Naples parking functions: apply the classical increasing-rearrangement criterion to $T(\alpha)$.
The bijection in Theorem 1.3 means decreasing $k$-Naples parking functions are counted by the number of signature Dyck paths with signature $(k+1,2,\ldots,2,1,\ldots,1)$, matching the known sequences listed for $k=1,2,3$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The recursion's shape mirrors the set-partition and labeled-forest recurrences, so one could extract asymptotic growth rates for $|PF_{n,k}|$ from its dominant summand; the paper does not carry this out.
The map $T$ may generalize to $k>1$ by lowering entries that have already appeared in a block of size $k$; if such a map exists, it would give membership tests for all $k$, not just $k=1$.
The experimental observation that the largest gain $|PF_{n,k}\setminus PF_{n,k-1}|$ occurs at $k=1$ suggests a stronger statement: the incremental benefit of backward flexibility is greatest for the very first step, and a proof would likely use the recursion to bound successive differences.
Since $k$-lattice paths with this signature have known counts only for small $k$, the bijection points to a family of lattice-path counts whose uniform enumeration is open; deriving it would also enumerate decreasing $k$-Naples parking functions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The new object here is the k-Naples parking function, which interpolates between classical parking functions (k=0) and all preference lists (k≥n-1). That is a natural idea, and the paper gives a recursive formula (Theorem 1.1), a characterization for k=1 (Theorem 1.2), and a bijection with k-lattice paths (Theorem 1.3). The recursion and the lattice path bijection look right to me; the small cases in Table 1 check out, and the Pollak-style argument for contained parking functions (Lemma 3.4) is a nice adaptation.

The main soft spot is the proof of Theorem 1.1. The block decomposition—split cars into those parking left of reserved spot i+1 and those parking right—is correct in substance. A car with preference east of i+1 would park in i+1 before ever reaching the left block, because i+1 stays empty; so left-block preferences are indeed exactly PF_{i,k}. Similarly, right-block cars must have preferences at least i+2, and after shifting they are exactly the contained parking functions B_{n-i,k}. But the paper states both identifications in a sentence apiece and never proves the independence. A referee should ask for a lemma that pins down these two equivalences. It's a gap in exposition, not a fatal error—the recursion reproduces the table and the reasoning is recoverable—but it is load-bearing and needs to be written out.

Theorem 1.2 has a more serious proof problem. The map T is a nice idea, and examples support the characterization, but the proof that α being a Naples parking function implies T(α) is a classical parking function does not go through. It argues that each car can park because "there must be an empty spot between a_i-1 and n," but the occupancy pattern in the classical process with preferences T(α) is not the same as in the Naples process with α. Previous cars park differently, so the same spot may not be available. The converse has the same issue. This needs a real proof, probably by induction on cars or via the increasing rearrangement condition.

The lattice path bijection (Theorem 1.3) is more solid. The proof is a bit informal, especially the "most optimal" contradiction, but the condition a_i ≤ min(n, n+k+1-i) is the right one and the construction is clear.

Who is this for? Enumerative combinatorists working on parking functions and lattice paths. It doesn't resolve any long-open problem, but it gives a new family with a computable recursion and a connection to signature Dyck paths. I'd send it to a serious referee, with the expectation of heavy revision on Sections 3 and 4.

Best, [Your name]

Referee Report

3 major / 5 minor

Summary. The paper introduces k-Naples parking functions, in which a car whose preferred spot is occupied may first back up up to k spaces before continuing east. The authors prove three main results: a recursion (Theorem 1.1) for the number |PF_{n+1,k}| of k-Naples parking functions, a characterization of the k=1 case via a map T to classical parking functions (Theorem 1.2), and a bijection between decreasing k-Naples parking functions and k-lattice paths (Theorem 1.3). The recursion is checked against a table of values computed by code posted on GitHub. The paper also gives a Pollak-style counting argument for a set of 'contained' parking functions and a bijection for the top two diagonal cases. The overall narrative is clear and the conjectured family is natural, but the proof of the main recursion and the proof of the k=1 characterization contain gaps that need to be repaired before the claims are fully established.

Significance. If the gaps are repaired, the paper would be a worthwhile contribution: the recursion genuinely interpolates between the classical parking-function count (n+1)^{n-1} at k=0 and the full set n^n at k>=n-1, the circular-counting idea is elegant, and the lattice-path bijection gives a new enumerative handle on decreasing k-Naples parking functions. The computational table and the GitHub code are a useful check and a reproducible complement to the proofs. The paper also identifies several natural open problems and gives explicit OEIS connections. The main deficits are in proof completeness rather than in the plausibility of the results.

major comments (3)

[Section 3, Eq. (2), proof of Theorem 1.1] The proof asserts, after fixing the subset S of cars that park to the left of spot i+1, that the number of preference assignments to S is |PF_{i,k}| and that the number of assignments to the complement is |B_{n-i,k}|, without proving either factorization. For S, one must show that no car in S can have a preference east of spot i+1, and that the induced parking process on spots 1 through i is exactly the k-Naples process on i cars; because spot i+1 is empty until the last car, this is plausible but not automatic. For the complement, one must show that after relabeling spots i+2 through n+1 as 1 through n-i, the condition 'these cars avoid parking in spot i+1' is exactly the contained-parking condition in Definition 3.2. These two factors are the core of the recursion, so Eq. (2) is not established by the present argument.
[Section 3, Definition 3.2 and Lemma 3.4] Definition 3.2 is not well-formed as written: the condition 'if cars c1,...,c_{i-1} have already filled spaces 1,...,a_i, then there is no car c_i with a parking preference 1 <= a_i <= k' uses i both as a running index and as the length parameter, and does not clearly define the set B_{n,k}. Lemma 3.4 then states |B_{n,k}|=(n+1)^{n-1}; because the defining set is ambiguous, this count is not verifiable as stated. The proof of Lemma 3.4 also switches inconsistently between 'spot 0' and 'spot n+1' in the circular construction, and it asserts the equivalence between B_{n,k} and 'the cars park leaving the extra spot empty' without proof. This is load-bearing, since Eq. (2) uses the value |B_{n-i,k}|=(n-i+1)^{n-i-1}.
[Section 4, proof of Theorem 1.2] The forward direction argues that if car c_i can park under the Naples rule, then there is an empty spot q between a_i-1 and n, and concludes that c_i can park under the classical rule with preference b_i in T(alpha). This ignores that the set of occupied spots when c_i arrives is different in the two processes: earlier cars may occupy different spots under the Naples rule than under the classical rule, so q need not be empty in the T(alpha) process. The converse direction has the same problem in reverse. The proof needs an invariant relating the partial parking configuration of alpha to that of T(alpha), or a different argument. As written, the characterization of Naples parking functions is not established.

minor comments (5)

[Example 4.2] The text says 'tau(a1)=a1=4', but for alpha=(2,4,4,1) the first entry is 2, so the example should read tau(a1)=2.
[Lemma 3.4] The paragraph following Lemma 3.4 refers to 'the proof of Theorem 3.4', but the result is numbered Lemma 3.4.
[Lemma 2.2] The proof only analyzes the last car. To conclude beta in PF_{n,n-2}, it should also justify that when b_n != n, no earlier car ever uses the full n-1 backward steps; otherwise the earlier cars' parking positions could change when k is reduced. This is likely repairable but should be stated.
[Section 5, proof of Theorem 1.3] The equivalence between the path condition and the inequality a_i <= min(n,n+k+1-i) is stated without explaining the coordinate convention; the proof checks the bound at the right endpoint of each east step, and this convention should be made explicit in Definition 5.3.
[General presentation] There are several minor typographical issues, including 'Cellabos' for 'Ceballos' and an incomplete author listing in reference [9]; these should be corrected in revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main recursion is a genuine recurrence and the auxiliary Pollak-style counting is independent.

full rationale

The main result (Theorem 1.1) is a recursive formula for |PF_{n+1,k}| in terms of the smaller quantities |PF_{i,k}| and (n-i+1)^{n-i-1}; this is an ordinary recurrence, not a prediction that is forced by its own inputs. The factor |PF_{i,k}| is a lower-order instance of the object being counted, and the factor (n-i+1)^{n-i-1} is derived independently in Lemma 3.4 via a Pollak-style circular parking argument, not by assuming the theorem. The paper does not fit any parameter to data and then rename that fit as a prediction; the GitHub-computed table is presented as computational evidence and is not used to prove the main formula. The proof of Theorem 1.1 does contain an asserted block factorization: it says without a fully detailed proof that the preferences of the cars to the left of spot i+1 are counted by |PF_{i,k}| and those to the right by |B_{n-i,k}|. That is a potential gap or a point needing a stronger independence lemma, but it is not circularity: the key counting factors are independent of the target recursion and would need independent justification rather than being definitions of the result. The proofs of Theorems 1.2 and 1.3 are self-contained combinatorial arguments, and the connection to signature Dyck paths is a translation of the newly introduced k-lattice paths, not a repackaging of a known result as a derivation. There are no load-bearing self-citations: the cited classical parking function results, Pollak technique, and signature Dyck path framework are external and not supplied by the present authors. Overall, the derivation chain is self-contained and no step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No free parameters are fitted; k and n are variables. The paper relies on standard parking function characterizations and standard counting identities. The introduced objects (k-Naples functions, contained functions, k-lattice paths) are definitions, not unexplained entities.

assumptions (2)

standard math A preference tuple is a classical parking function iff its increasing rearrangement b satisfies b_i <= i.
Used in Section 4 to evaluate T(alpha); cited to Konheim and Weiss [5].
standard math A decreasing preference tuple (a1 >= ... >= an) is a classical parking function iff a_i <= n-i+1 for all i.
Used in Section 5 to relate decreasing parking functions to Dyck paths; follows from the same characterization.

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Pith. "Pith review of A Generalization of Parking Functions Allowing Backward Movement." pith.science (2026). https://pith.science/paper/J2DSOABG

@misc{pith2026190807658,
  author       = {Pith},
  title        = {Pith review of: A Generalization of Parking Functions Allowing Backward Movement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J2DSOABG}},
  note         = {Machine review of arXiv:1908.07658}
}

abstract

Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the $n$-tuple containing the cars' parking preferences a parking function. In this paper, we introduce a generalization of the parking rule allowing cars whose preferred space is taken to first proceed up to $k$ spaces west of their preferred spot to park before proceeding east if all of those $k$ spaces are occupied. We call parking preferences which allow all cars to park under this new parking rule $k$-Naples parking functions of length $n$. This generalization gives a natural interpolation between classical parking functions, the case when $k=0$, and all $n$-tuples of positive integers $1$ to $n$, the case when $k\geq n-1$. Our main result provides a recursive formula for counting $k$-Naples parking functions of length $n$. We also give a characterization for the $k=1$ case by introducing a new function that maps $1$-Naples parking functions to classical parking functions, i.e. $0$-Naples parking functions. Lastly, we present a bijection between $k$-Naples parking functions of length $n$ whose entries are in weakly decreasing order and a family of signature Dyck paths.

Figures

Figures reproduced from arXiv: 1908.07658 by the authors.

**Figure 1.** Parking function illustration. A parking preference of length n is an n-tuple of integers in [n] where the i-th component corresponds to the preferred parking spot of car ci . We denote the set of parking preferences of length n as P Pn. Note that |P Pn| = n n . For a parking preference α = (a1, . . . , an) ∈ P Pn, we establish the following parking rule: for all 1 ≤ i ≤ n, ci starts at parking space 1 and drives to… view at source ↗

**Figure 2.** Plots for |P F∗ n,k| for varying values of n and with 1 ≤ k ≤ n. The scale of the y-axis is scaled by a factor of 1034, 1083, 10139, and 10198, when n = 25, 50, 75, and 100, respectively. Given this observation, Naples parking functions are of particular interest. Our next main result gives a necessary and sufficient condition to characterizing Naples parking functions. Theorem 1.2. Fix n ∈ N. Let α = (a1, a2, . . .… view at source ↗

**Figure 3.** Illustration of order in which cars with preference (1, 3, 3, 2) park under the classical parking rule (left) and under the Naples parking rule (right). We observe that for any parking preference of length n there is a maximum of n − 1 steps backward that a car can take from its preferred parking space. Moreover, if each car can take up to n − 1 steps backwards then each car is able to check each of the n spaces and… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Labeling the parking spaces in two distinct ways. Since α ∈ P Fn−1, the n − 1 cars park in the red labeled spaces 1 to n − 1, moving from east to west, or the in black labeled spaces 2 to n, moving west to east. Moreover, car di with parking preference ψ(ai) = n + 1 − …

**Figure 5.** Figure 5: Parking position of cars with parking preference α = (4, 4, 2, 3). If β = (4, 2, 2, 2), then the cars park as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Parking position of cars with parking preference α = (4, 2, 2, 2) Hence β = (4, 2, 2, 2) ∈/ B4,2, because c4 was able to look back past spot 1 and park in spot 0, leaving spot 3 empty. Thus, the cars’ final parking positions are not contained in spots 1 through 4. Howe…

**Figure 7.** Figure 7: Dyck path corresponding to α = (4, 3, 3, 1, 1). We now consider a generalization of Dyck paths, which we call k-lattice paths. Definition 5.3. If n, k ∈ N with 0 ≤ k ≤ n − 1, then a k-lattice path of length 2n is a lattice path from (0, n) to (n, 0) consisting of n ste…

**Figure 8.** Figure 8: The ribbon corresponding to the signature s = (3, 2, 5, 1, 1), and two lattice paths: one an s-Dyck path (blue path) and that is not an s-Dyck path (red path). In our work, we consider a horizontal reflection of s-Dyck paths so that our paths are decreasing, rather tha…

**Figure 9.** Figure 9: Illustrating the possible locations for 3-lattice paths of length 12, which begin at (0, 6), end at (6, 0), and must lie below the red line given by y = 6 − x + 3. This corresponds to s-Dyck paths of length 14, which begin at (0, 7), end at (7, 0), and must lie on or b…

**Figure 10.** Figure 10: Illustrating the parking order for the 3-Naples parking function (7, 7, 7, 7, 5, 2, 2). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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work page 2019
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[9]

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2015

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alyson Baumgardner, The naples parking function, Honors Contract-Graph Theory, Florida Gulf Coast University, 2019.↑2

work page 2019

[2] [2]

Cameron, Daniel Johannsen, Thomas Prellberg, and Pascal Schweitzer, Counting defective parking func- tions, Electron

Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, and Pascal Schweitzer, Counting defective parking func- tions, Electron. J. Combin. 15 (2008), no. 1, Research Paper 92, 15. MR2426155 ↑2

work page 2008

[3] [3]

Gonz´ alez D’Le´ on,Signature catalan combinatorics , Journal of Combinatorics 10 (2019), 725–773.↑4, 10, 12

Cesar Ceballos and Rafael S. Gonz´ alez D’Le´ on,Signature catalan combinatorics , Journal of Combinatorics 10 (2019), 725–773.↑4, 10, 12

work page 2019

[4] [4]

10 (1974), 10–22

Dominique Foata and John Riordan, Mappings of acyclic and parking functions , Aequationes Math. 10 (1974), 10–22. MR0335294 ↑7

work page 1974

[5] [5]

Konheim and Benjamin Weiss, An occupancy discipline and applications , Siam Journal on Applied Mathematics - SIAMAM 14 (196611).↑1, 2, 6

Alan G. Konheim and Benjamin Weiss, An occupancy discipline and applications , Siam Journal on Applied Mathematics - SIAMAM 14 (196611).↑1, 2, 6

work page

[6] [6]

Ronald Pyke, The supremum and inﬁmum of the poisson process, Ann. Math. Statist. 30 (195906), no. 2, 568–576. ↑1

work page

[7] [7]

Paul R. F. Schumacher, Descents in parking functions, J. Integer Seq. 21 (2018), no. 2, Art. 18.2.3, 8. MR3779772 ↑8

work page 2018

[8] [8]

Yan, Generalized parking functions, tree inversions, and multicolored graphs , Adv

Catherine H. Yan, Generalized parking functions, tree inversions, and multicolored graphs , Adv. in Appl. Math. 27 (2001), no. 2-3, 641–670. Special issue in honor of Dominique Foata’s 65th birthday (Philadelphia, PA, 2000). MR1868985↑2

work page 2001