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Distribution of new statistics of parking functions and their generalizations

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Pith's one-line read Two parking-function statistics have one exact joint counting formula.

desk verdict A careful enumerative paper that resolves a known equidistribution with a new statistic and a mostly sound transfer to forests; the only real weakness is the deferred BFS bijection proof. read the letter →

arxiv 2507.20495 v1 pith:5UTC36NH submitted 2025-07-28 math.CO

Stephan Wagner , Catherine H. Yan , Mei Yin This is my paper · ORCID
classification math.CO MSC 05A1505A1960C05
keywords parkingfunctionslabeledforestsbreadth-firstsearchleadingelementslevelsetstatisticjointdistribution(ab)-parkingcoloredtrees
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

This paper derives the exact joint distribution of two statistics on parking functions: slev, the number of cars whose preferred spot lies in the low range {1,...,n-m+1}, and lel, the number of cars that prefer the same spot as the first car. The main theorem gives a closed multinomial-type formula for the number of parking functions with slev=s and lel=t, together with its two-variable generating function. The same pair of statistics becomes, through a breadth-first-search bijection, the total root degree and the degree of vertex 1's parent in rooted forests, so the enumerative results for parking functions and forests feed each other. In the classical case m=n, this explains the previously observed but mysterious symmetric equidistribution between the ones statistic and the leading-elements statistic, and it provides explicit bijections that swap the two statistics. The same machinery extends to (a,b)-parking functions and colored trees, where the symmetry reappears at block level.

What carries the argument

The key object is the generalized breadth-first-search bijection between parking functions PF(m,n) and rooted forests F(m,n), an extension of the classical construction for m=n. A parking function is encoded by its specification s(π), the counts #k(π) of each preferred spot, together with an order permutation τπ that records the position of each entry in the non-decreasing rearrangement; the same pair, subject to balance and compatibility conditions in the set C(m,n), encodes a rooted forest's degree sequence and BFS ordering. The bijection works because π_i=j exactly when vertex i is a child of the j-th vertex of the forest in BFS order. Consequently the statistic slev is exactly the total degree of the roots, lel is the degree of the parent of vertex 1, and every count of a preference value becomes the number of children of the corresponding BFS vertex. Counting forests by these two degree statistics, with the parent of vertex 1 either a root or not, reproduces the parking-function formula and supplies the bijective explanations.

What would settle it

For a small pair such as m=3, n=5, list all 108 parking functions in PF(3,5), tally (slev, lel), and compare the counts with Theorem 2.4; then list all 108 rooted forests in F(3,5), tally (deg(0), deg(p)), and compare with Proposition 4.2 and with the BFS images of the parking functions. Any mismatch in the two tables, or any feasible pair in C(3,5) that does not correspond to exactly one parking function and one forest, would refute the central claim.

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Extended reading notes

Core claim

The central claim is Theorem 2.4: for s,t≥1, the number of π∈PF(m,n) with slev(π)=s and lel(π)=t is $$inom{m-2}{s-1,t-1,m-s-t}(n-m+1)^s(m-1)^{m-s-t+1}+inom{m-1}{t-1,s-t,m-s}s(n-m+1)(n-m)^{s-t}$m^{{m-s-1}}$.$$ The proof splits according to whether the first car prefers a spot in the low range {1,...,n-m+1} or not, using a decomposition that isolates the entries in that range and reduces the rest to a smaller parking function. Summing the formula yields the bivariate generating function in Corollary 2.5. Via the generalized breadth-first-search correspondence, these counts transfer to rooted forests, where slev becomes the total root degree deg(0) and lel becomes deg(p), the number of children of the parent of vertex 1. This transfer gives a forest-theoretic explanation of the symmetric equidistribution of the classical ones and leading-elements statistics, with explicit involutions that swap the two statistics while preserving the remaining structure.

Load-bearing premise

The load-bearing assumption is that the generalized breadth-first-search construction is a bijection for every (m,n), since the paper defers the technical verification to an earlier argument and the statistic transfer collapses if some feasible pair is not realized one-to-one.

Editorial extensions

If this is right

  • The bivariate generating function of Corollary 2.5 contains the full joint distribution; setting one variable to 1 yields the closed univariate generating functions for lel and slev separately.
  • At m=n, slev is the classical ones statistic, so the formula reproves the symmetric equidistribution of ones and leading elements and gives it a forest interpretation.
  • Every parking-function enumeration in the paper transfers to an enumeration of rooted forests by the pair (deg(0), deg(p)), so results in either setting apply to the other.
  • The involutions θ and ρ provide explicit bijections that swap the two statistics; ρ preserves the reduced preference partition and thereby refines the symmetry to subsets of parking functions defined by fixed set partitions.
  • For (a,b)-parking functions, analogous generating functions hold, and for (1,b) and (k,k) cases the same symmetries and bijections extend to colored trees, giving block-level refinements.

Reading between the lines

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

  • The pair (lel, slev) may be asymptotically independent in the regime m=cn, since both marginal generating functions factor into independent Bernoulli sums; the paper proves only the univariate Poisson and normal limits, so this joint limit is not claimed.
  • The decomposition underlying Lemma 2.2 suggests a recursive construction of uniformly random parking functions by first choosing the low-range entries and then a smaller parking function, which could be exploited for simulation or for higher-dimensional joint statistics.
  • The same feasible-pair encoding may yield exact joint distributions for refined statistics such as the individual counts #k(π) beyond the (a,b)-parking-function case, where the paper gives only partial refinements.
  • The bijections θ and ρ, defined for classical parking functions and forests, likely adapt to swap the root degree with the degree of the parent of any fixed vertex i, not only vertex 1, since the degree symmetry is argued by vertex symmetry.
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Editorial analysis

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

Referee Report

2 major / 4 minor

Summary. The paper studies two statistics on generalized parking functions PF(m,n): slev, the number of entries in {1,...,n-m+1}, and lel, the multiplicity of the first entry. Theorem 2.4 gives a closed formula for the joint distribution of (slev, lel), Corollary 2.5 gives the bivariate generating function, and specializations yield the marginal distributions. The authors then introduce a generalized breadth-first-search bijection between PF(m,n) and rooted forests F(m,n), and use it to transfer (slev, lel) to (deg(0), deg(p)), the total root degree and the degree of the parent of vertex 1. For m=n this is used to explain and give an explicit bijection for the equidistribution of ones and lel observed by Stanley and Yin. The paper also proves a direct forest enumeration (Proposition 4.2), gives two involutions θ and ρ on labeled trees, derives limit laws for the statistics, and extends the results to (a,b)-parking functions and colored trees.

Significance. If the generalized BFS bijection in Section 3 is supplied, the paper gives a satisfying explanation of the Stanley-Yin equidistribution, an explicit joint distribution theorem with a compact generating function, and interesting refinements via the involutions θ and ρ. Theorem 2.4 and the independent forest count in Proposition 4.2 appear correct, and the decomposition lemma (Lemma 2.2) is an elegant tool. The asymptotic results are straightforward consequences of the generating functions but are cleanly stated. Overall the contributions are valuable and within the scope of math.CO, provided the transfer bijection and the (a,b)-parking-function decomposition are made fully correct.

major comments (2)
  1. [Section 3] Section 3 asserts, but does not prove, that the generalized breadth-first-search construction is a bijection between PF(m,n) and F(m,n). After defining t(f) and σ_f, the text says the construction may be similarly argued as in [5] with minor adaptations, and that technical details will not be given. This assertion is load-bearing: Theorem 3.1, Theorem 4.1, Corollaries 5.2 and 5.4, and the bijective explanations in Sections 6 and 8 all rely on the fact that the feasible pairs (s(π), τ_π) for parking functions coincide with the feasible pairs (t(f), σ_f) for forests under the same set C(m,n). A proof, or a precise citation covering the generalized (m,n) case, is needed. The queue-balance condition appears to match the parking-function inequality (1), so I do not believe the assertion is false, but as written this is a genuine gap in the exposition.
  2. [Section 7, Lemma 7.2] Lemma 7.2 contains an incorrect reconstruction identity for (a,b)-parking functions. The text states that π_{j_i}=b(\tildeπ_i−1)+α_i with α_i≡π_{j_i} mod b. This identity misses the additive constant a. For example, with a=3, b=2 and π_{j_i}=4, we have \tildeπ_i=⌈(4−3)/2⌉=1, and the only α_i∈[2] congruent to 4 modulo 2 is α_i=2, but then b(\tildeπ_i−1)+α_i=2, not 4. The correct identity is π_{j_i}=a+b(\tildeπ_i−1)+α_i, where α_i∈[b] is the remainder of π_{j_i}−a in [0,b−1], equivalently α_i≡π_{j_i}−a (mod b). The counting arguments in Propositions 7.3, 7.5 and 7.10 use the number b^{m−s} of possible α's and the independence of α_i, which remain valid after this correction, but Lemma 7.2 as stated must be revised.
minor comments (4)
  1. [Lemma 2.2 and Lemma 7.2] The notation 'π_{j_1}<π_{j_2}<...' is inaccurate because entries outside the level set need not be distinct; for example π=(4,4,3) is a parking function in PF(3,5) with two equal entries outside {1,2,3}. The intended meaning is a non-decreasing ordering.
  2. [Section 3] The set C(m,n) of feasible pairs is never defined in the paper; the reader must consult [8, Section 2.2]. Since the proof of the BFS bijection depends on the exact feasibility conditions, stating at least the conditions explicitly would make the argument verifiable.
  3. [Start of Section 7] The paragraph explaining the reduction to the case m=n for (a,b)-parking functions is easy to misread. For a u-parking function of length m with n available spots, the relevant condition is λ_i≤u_{i+n-m}; the equivalence with PF(a+(n-m)b,b,m) is correct, but the wording should make this indexing explicit.
  4. [Section 7, Proposition 7.9] Proposition 7.9 is stated without proof, although it follows from Corollaries 7.7 and 7.8 by the same argument as Proposition 5.1. A one-sentence proof or a reference to the analogous argument would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main joint distribution is derived from Pollak's circle argument and an elementary decomposition, while self-citations are contextual or re-derived.

full rationale

The central enumeration (Theorem 2.4 and Corollary 2.5) is self-contained: Lemma 2.2 gives an equivalence with a smaller parking function, and the proof then builds on Pollak's circle argument, not on the theorem being proved. Corollaries 2.6 and 2.7 re-derive the Stanley–Yin equidistribution inside the paper rather than importing it from [17]. The forest-side formula is independently established in Proposition 4.2 by a direct combinatorial construction, so it does not depend on the parking-to-forest bijection. The Section 3 BFS bijection is asserted with proof deferred to Foata–Riordan [5] and feasible-pair definitions cited from Kenyon–Yin [8]; this is an omitted proof or exposition gap, not a circular reduction, since the explicit forward and inverse formulas are given and the main enumeration does not rely on the bijection. The self-citations to [8], [17], [18], and [19] are either contextual, re-derived, or to published prior constructions; none is used to substitute for a derivation of the target results. Proposition 7.9 is stated without proof, but that is an omitted detail rather than a circular step. No fitted parameter is renamed as a prediction, and no uniqueness theorem is invoked from the authors' prior work.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters appear anywhere; all results are exact closed-form counts and generating functions. The statistics slev and lel are definitions on existing structures, not auxiliary entities summoned to make a derivation work, and the involutions theta, rho, rho1,b, and rhok,k are maps rather than postulates. The axioms are classical background: the parking-function characterization, Pollak's circle argument, Cayley's forest count, and the BFS and colored-tree bijections inherited from [5], [8], and [19]. The ledger is therefore clean: the paper adds exact formulas and bijections on top of standard inputs.

assumptions (5)
  • domain assumption Pollak's circle argument: in the cyclic shift action on (Z/(n+1)Z)^m, every coset contains exactly n-m+1 parking functions in PF(m,n).
    Used throughout Section 2 (Lemma 2.1, Theorem 2.4) and in Section 7 for (a,b)-parking functions; classical, recounted in [5], [12], and [16].
  • standard math Parking function characterization (1): pi in PF(m,n) iff #{k: pi_k <= i} >= m-n+i for i = n-m+1, ..., n.
    Elementary pigeonhole characterization used as the basis of Lemma 2.2 and Lemma 7.2.
  • standard math Cayley-style forest count: the number of rooted forests on [a] with b specified roots is b * a^(a-b-1).
    Load-bearing input in Proposition 4.2's direct combinatorial proof, cited to Stanley [15].
  • domain assumption The breadth-first-search bijection between PF(m,n) and F(m,n), and its colored-tree generalization for (a,b)-parking functions.
    Section 3 invokes the Foata-Riordan correspondence [5] with 'minor adaptations'; Section 8 invokes Yan's (a,b)-colored tree bijection [19]. The paper does not reproduce the full feasibility conditions from [8, Section 2.2].
  • domain assumption General u-parking function characterization: pi is a u-parking function iff #{k: pi_k <= u_i} >= i for all i, equivalently the sorted entries satisfy lambda_i <= u_i.
    Used in Lemma 7.2 and Proposition 7.3 for (a,b)-parking functions, following the standard theory in [20].

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Pith. "Pith review of Distribution of new statistics of parking functions and their generalizations." pith.science (2026). https://pith.science/paper/5UTC36NH

@misc{pith2026250720495,
  author       = {Pith},
  title        = {Pith review of: Distribution of new statistics of parking functions and their generalizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5UTC36NH}},
  note         = {Machine review of arXiv:2507.20495}
}
read the original abstract

In this paper we present new results on the enumeration of parking functions and labeled forests. We introduce new statistics on parking functions, which are then extended to labeled forests via bijective correspondences. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics. Extensions of our techniques are discussed, including joint distribution on further refinement of these new statistics.

Figures

Figures reproduced from arXiv: 2507.20495 by the authors.

Figure 1
Figure 1. Rooted spanning forest. We read the vertices of the forest in breadth first search (BFS) order. That is, read root vertices in order first, then all vertices at level 1 (children of a root), then those at level 2 (distance 2 from a root), and so on, where vertices at a given level are naturally ordered in order of increasing predecessor, and, if they have the same predecessor, increasing order. See [20, Section 1.2.… view at source ↗
Figure 2
Figure 2. A bijective map between deg(0) and deg(p) (illustration). See [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. An example of the map θ. Proposition 4.2. Let s, t ≥ 1. We have #{F ∈ F(m, n): deg(0) = s and deg(p) = t} =  m − 2 s − 1, t − 1, m − s − t  (n − m + 1)s (m − 1)m−s−t+1 +  m − 1 t − 1, s − t, m − s  s(n − m + 1)(n − m) s−tmm−s−1 . Proof. As a starting point, we recall the well-known result that the number of rooted forests with vertex set [a] and b components whose root labels are given is baa−b−1 (see [15]). Thu… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Illustration of the procedure: the rooted forest (top) with the distinguished vertex p indicated by a thick node, and the final tree (bottom). Case 2. The parent p of vertex 1 is one of the roots. Again, there is a unique way to construct all these forests: • Select a …
Figure 5
Figure 5. Figure 5: The involution ρ swaps deg(0) and deg(p) (illustration). For any vertex q ̸= 0 or p, let C(q) be the set of children of q. Then the nonempty sets of the form C(q), together with K = C(0) ∪ C(p), form a set partition of [n], which is preserved under ρ. In terms of parki…
Figure 6
Figure 6. Figure 6: An example of the map ρ. The preference partition of π is {{1}, {2}, {3, 8}, {4, 6, 7}, {5}, {9}}, and K(π) = {1, 4, 6, 7}. One sees that K(ˆπ) = {4, 1, 6, 7} = K(π), and π and ˆπ have the same reduced preference partition. Theorem 6.1 has some interesting consequences…
Figure 7
Figure 7. Figure 7: An (a, b)-colored tree T with n = 8, a = 3 and b = 2. are of the same color, in increasing order. For example, the (a, b)-colored tree in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: An example of a (2, 2)-parking function π and the corresponding (2, 2)-colored tree T. We explain the above concepts with an example of π = (10, 3, 2, 9, 3, 1) ∈ PF(2, 2, 6), which corresponds to the (2, 2)-colored tree on the left of [PITH_FULL_IMAGE:figures/full_fig…

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