arXiv:2603.24886v1

Bijectivity of a generalized Pak-Stanley labeling

Olivier Bernardi, Neha Goregaokar

math.CO

Paper metadata

Original submission: March 26, 2026
Claimed MSC
arXiv categories: math.CO
Paper license: arXiv non-exclusive distribution license ↗
Comments: 14 pages, 6 figures. Extended abstract accepted to FPSAC 2026

Abstract

The Pak-Stanley labeling is a bijection between the regions of the $m$ -Shi arrangement and the $m$ -parking functions. Mazin generalized this labeling to every deformation of the braid arrangement and proved that this labeling is always surjective onto a set of directed multigraph parking functions. We provide a right inverse to the generalized Pak-Stanley labeling, and identify a class $\mathcal{C}$ of arrangements for which this labeling is bijective. The class $\mathcal{C}$ includes the multi-Shi arrangements and the multi-Catalan arrangements. We also show that the arrangements in $\mathcal{C}$ are the only transitive arrangements for which the generalized Pak-Stanley labeling is bijective.

Dependence graphs

Proof lineage

Bijectivity of a generalized Pak-Stanley labeling

A statement-restricted dependence graph for the right-inverse construction, the bijectivity criterion for (m, epsilon)-arrangements, and the necessity theorem inside the transitive class. Only external results that can affect those marked claims are included. Historical comparisons, alternative proofs, and citations used only for context are excluded; expansion stops at published books and monographs.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 23, 2026

01Statements5 reported findingsContains wrong statements

The right-inverse theorem and the sufficient bijectivity theorem for $(m,\varepsilon)$ -arrangements are correct after the verified local repairs. Theorem 5.1 is false in its stated positive- $n$ scope: for $n=2$ and $S_{1,2}=\{2,7\}$ , the arrangement is transitive and its three regions are bijectively labeled, but it is not an $(m,\varepsilon)$ -arrangement. The exhaustive $n=3$ tests still support a possible corrected classification in a narrower scope, although the general converse sketches remain incomplete. Several foundational definitions and the $m=0$ queue boundary also require explicit repairs.

Definitions 2.1–2.2Unsupported as written · verified repair

The parking-function domain and the off-diagonal arc set must be stated

PDF page 4 · Definitions 2.1–2.2 · arXiv:2603.24886v1

Definition 2.2 calls $p=(p_1,\ldots,p_n)$ only a sequence. With no requirement that its entries be nonnegative integers, every sufficiently negative integer or real vector satisfies the displayed upper inequalities, while the algorithm of Section 3 can start with neither a zero nor an open-list entry and return an incomplete word. The intended and verified repair is $p\in\mathbb{Z}_{\geq0}^n$ . Definition 2.1 also quantifies over all $i,j\in[n]$ , although $S^+_{i,i}$ is never defined; replacing this by $i\ne j$ gives the intended loop-free directed multigraph. Under these repairs, the sink arc makes each parking-function coordinate finite and the Section 3 algorithm has the required domain.

Exact arXiv version 1 ↗

Definitions 1.3 and 2.7–2.8Ill-defined as written · verified conventions

Index and empty-set conventions are missing from the main scope

PDF pages 3 and 5–6 · Definition 1.3, Definition 2.7, Definition 2.8, and the setup of Section 3 · arXiv:2603.24886v1

Definition 2.7 quantifies $i,j,s,t$ but uses an unbound index $k$ ; Definitions 2.7, 4.3, and 4.5 also range into diagonal symbols $S^+_{i,i}$ that the notation never defines. Bernardi's imported definition uses distinct indices, and restricting the displayed conditions to the pairwise-distinct indices actually occurring in their $S^+$ terms is the verified repair. In addition, $m=\max\bigcup S^+_{i,j}$ is undefined for the empty arrangement, although every tuple of finite sets is allowed. Taking this maximum to be zero and declaring $[a;b]=\varnothing$ for $b<a$ makes Definition 1.3 represent the empty boundary with $m_k=0$ and all relevant $\varepsilon_{i,k}=1$ . It does not by itself repair Section 3: when $m=0$ , Case 1 must append $k$ to the open list only if $m>0$ , or Case 2 emits a nonexistent letter $\alpha_k^{(1)}$ . A local-maximality clause with no qualifying adjacency must likewise be read vacuously.

Bernardi, exact version of record ↗

Theorem 3.2Correct after verified local repairs

The right-inverse identity survives the definition repairs

PDF pages 6–8 · Lemmas 3.1 and 3.5 and Theorem 3.2 · arXiv:2603.24886v1

For a nonnegative directed-graph parking function, the zero/queue procedure cannot terminate with a positive coordinate: applying the parking inequality to the remaining positive index set gives the contradiction printed in Lemma 3.1. First-in-first-out processing preserves the sketch order, and the coordinate invariant in Lemma 3.5 counts exactly the source letters preceding each zero-letter. Evaluating that invariant when the zero-letter is emitted proves the right-inverse identity coordinate by coordinate. Two boundary repairs are necessary: the printed terminal value $-m$ must be $-(m+1)$ , and Case 1 must enqueue $k$ after $\alpha_k^{(0)}$ only when $m>0$ . Without the latter repair, the empty $n=2,m=0$ case with $p=(0,0)$ emits a nonexistent level-one letter. With both repairs, exhaustive checks covered 742,656 parking-function inputs across all 33,280 three-index arrangements with $m=1$ or $m=2$ , and a dedicated $m=0$ regression check, with no failure.

Exact arXiv version 1 ↗

Theorem 4.1Correct after verified local repairs

The sufficient bijectivity theorem uses only the sound half of Lemma 4.4

PDF pages 8–11 · Theorem 4.1, Lemmas 4.2 and 4.6, and the forward half of Lemma 4.4 · arXiv:2603.24886v1

An $(m,\varepsilon)$ -arrangement directly satisfies Properties X and Y. The proof of Theorem 4.1 then uses Y only in the sound direction $Y\Longrightarrow M_S=L_S$ , uses Lemma 4.6 to obtain $N_S=M_S$ from X, and invokes Bernardi's surjection through Lemma 4.2. It does not use the ellipsis-based converse $M_S=L_S\Longrightarrow Y$ . After the off-diagonal index and empty-interval conventions are inserted and the Section 3 algorithm queues a zero-level index only when $m>0$ , each required implication is verified and the explicit inverse is the repaired Section 3 right inverse.

Exact arXiv version 1 ↗

Theorem 5.1Wrong as stated · explicit n=2 counterexample

The transitive classification is false in the stated positive-n scope

PDF pages 11–13 · Theorem 5.1 and Lemma 5.3 · arXiv:2603.24886v1

Let $n=2$ and $S_{1,2}=\{2,7\}$ , so $S^+_{1,2}=\{2,7\}$ and $S^+_{2,1}=\varnothing$ . Bernardi transitivity is vacuous because it quantifies pairwise-distinct triples. The arrangement consists of two parallel hyperplanes and has three regions. Its graph has two arcs $1\to2$ and one sink arc from each vertex, hence $\operatorname{Park}_S=\{(0,0),(1,0),(2,0)\}$ ; direct evaluation of $\lambda$ gives exactly these three distinct labels, so condition (a) holds. After the necessary distinct-index repair, Properties X and Y are also vacuous, so (c) holds. But an $(m,\varepsilon)$ -arrangement would require $S^+_{1,2}=[1;m_2-\varepsilon_{1,2}]$ , an initial interval, which $\{2,7\}$ is not. Thus (a) and (c) do not imply (d), disproving Theorem 5.1 in its stated scope. Exhaustive enumeration still found the claimed equivalences for all 512 arrangements with $n=3,m=1$ and all 32,768 with $n=3,m=2$ , but restricting to $n\geq3$ would only remove this counterexample; the incomplete converse sketches in Lemmas 4.4 and 5.3 would still need a general proof.

Exact arXiv version 1 ↗

02Proofs5 reported findingsContains incorrect or incomplete proofs

The Section 3 algorithm has a repairable off-by-one assertion and an $m=0$ queue error. The final reconstruction of the $(m,\varepsilon)$ parameters has an unbound witness and boundary error and, more decisively, cannot prove the false $n=2$ scope of Theorem 5.1. The converse sketch constructions in Lemmas 4.4 and 5.3 also omit the general extension and adjacency checks on which their contradictions depend; one displayed membership claim in Lemma 5.3 is false at shift zero. Finite exhaustive checks support the repaired claims at $n=3$ , but do not repair the theorem's stated scope or replace the missing arbitrary-size proof.

Lemma 3.1Incorrect as written · verified repair

The terminal queue coordinate is off by one

PDF pages 6–7 · first paragraph of the proof of Lemma 3.1 · arXiv:2603.24886v1

An index is decremented from zero to $-1$ when its zero-letter is emitted, and once more after each of its $m$ positive-level letters. Its terminal value is therefore $-(m+1)$ , not $-m$ as printed; Figure 4 itself ends at $-2$ when $m=1$ . Replacing $-m$ by $-(m+1)$ makes the ensuing conclusion exact: every letter at levels $0,1,\ldots,m$ appears once. There is a separate $m=0$ control-flow defect: Case 1 unconditionally appends the index to the open list, so Case 2 later emits $\alpha_k^{(1)}$ even though a $(0,n)$ -sketch has only zero-level letters. Appending only when $m>0$ repairs this boundary. The contradiction for the remaining positive set and the local-maximality check are then valid.

Exact arXiv version 1 ↗

Lemma 4.4Incomplete as written · unresolved in full generality

The two converse witnesses are specified only by ellipses

PDF page 9 · converse half of Lemma 4.4, Cases 1 and 2 · arXiv:2603.24886v1

Failure of Property Y is answered by displaying fragments such as a descending zero-letter block followed later by two prescribed adjacencies. In Case 2, the stated block before $\alpha_k^{(0)}$ includes every $\alpha_p^{(0)}$ with $p>k$ and $p\ne j$ ; because the case assumes $i>k$ , this already includes $\alpha_i^{(0)}$ , which the displayed word then writes a second time. The local repair is to exclude both $p=i$ and $p=j$ from that first block. Even after this correction, the proof does not define the omitted letters, verify all three sketch axioms, or check every zero-letter adjacency required for membership in $L_S$ . Those points are essential: an arbitrary word containing the fragment need not be a sketch. Exhaustive construction over every $(1,3)$ - and $(2,3)$ -sketch confirms the equivalence $Y\Longleftrightarrow M_S=L_S$ , and generic prefixes of the advertised kind had unique completions for $n=3,4$ and $m=2$ . This is strong support, but no general completion argument is supplied in version 1, so the converse remains unresolved as printed.

Exact arXiv version 1 ↗

Lemma 4.6Correct after verified index repair

The injectivity induction is valid after the distinct-index convention

PDF pages 9–10 · proof of Lemma 4.6 and equation (1) · arXiv:2603.24886v1

At the first differing position, two positive-level letters would force contradictory orders on their predecessors. Thus one word emits a zero-letter while the other delays it. Equality of the corresponding label coordinate excludes every contributing letter in the intervening segment, and Property X propagates the zero-level ordering backward through that segment to a contradiction. When the propagated index equals the fixed index, the needed alternative is immediate rather than an application of Property X. Restricting all $S^+$ symbols to unequal indices makes this implicit equality case and the displayed induction well-defined.

Exact arXiv version 1 ↗

Lemma 5.3Incorrect and incomplete · local repairs verified, general construction unresolved

The positive-shift witness contains a false membership claim and unproved completions

PDF pages 12–13 · second case in the proof of Lemma 5.3 · arXiv:2603.24886v1

From Property Y the proof claims $s-1\in S^+_{\ell,k}$ for every $\ell\ne j$ . This is false when $s=1$ and $\ell<k$ : zero is represented by the baseline clause in $\operatorname{Triple}_S$ , not by membership in $S^+_{\ell,k}$ . The exact repair is the weaker statement $(\ell,k,s-1)\in\operatorname{Triple}_S$ : for positive $s-1$ it follows from Y; at zero it follows either from the baseline order $\ell<k$ or, when $\ell>k$ , from Y. Also, $n^*=\max([n]\setminus\{i,j,k\})$ is undefined for $n=3$ and the corresponding block must be declared empty. These repairs justify the advertised boundary adjacencies. The proof still does not spell out the omitted suffixes, verify every local-maximality condition, or derive all equal label coordinates for arbitrary $n,m$ . The complete $n=3,m\le2$ enumeration found no failure, but the general witness verification remains unresolved.

Exact arXiv version 1 ↗

Theorem 5.1, (c) implies (d)Incorrect as written · theorem false at n=2

The parameter reconstruction has local errors and cannot cover n=2

PDF page 13 · final three paragraphs of the proof of Theorem 5.1 · arXiv:2603.24886v1

After fixing $k$ , the proof defines the least $s$ through $S^+_{i,k}$ without choosing or quantifying $i$ , and then uses that same free index throughout. A local repair is to choose a witness pair $(i,s)$ with globally minimal admissible $s$ , where $i\ne k$ and either $s>0$ or $i>k$ . In the branch where no set contains the boundary value, the printed choice $m_k=s-1$ is negative when $s=0$ ; the boundary repair is $m_k=s$ and $\varepsilon_{\ell,k}=1$ for every $\ell\ne k$ . The displayed monotonicity sentence is also backwards and should use the contrapositive of X, and $S_{\ell,k}$ is missing a plus sign. These corrections repair the printed local manipulations only where the argument has a third index. They cannot close the stated implication: for $n=2$ and $S_{1,2}=\{2,7\}$ , X and Y are vacuous while the shift set is not an initial interval. The proof's attempted contradiction after finding a later shift silently needs an index $j\ne i,k$ , which does not exist. A valid theorem must first narrow or otherwise amend its scope, and the remaining $n\geq3$ reconstruction must still be checked together with Lemmas 4.4 and 5.3.

Exact arXiv version 1 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

04Sources4 reported findingsContains incorrect or incomplete source use

Bernardi's 2018 paper is the only direct proof-critical non-book source. Its Section 8 results have the strength the focal proofs need, but the focal pinpoint for the region/sketch bijection names the wrong proposition, and the exact journal proof contains a defective inverse construction repaired only in arXiv:1604.06554v4. The focal transitivity definition also drops Bernardi's distinct-index quantifier. After those repairs, the inherited branch is mathematically applicable; its deeper proof-critical inputs terminate at Zaslavsky's monograph and Stanley's book.

Bernardi region/sketch correspondenceIncorrect or incomplete source use · verified repair

The focal pinpoint is wrong and the cited journal construction needs the later repair

PDF page 5 · Lemma 2.6 and citation [2, Proposition 8.1] · arXiv:2603.24886v1

Bernardi Proposition 8.1 is not the region/sketch bijection; it maps annotated sketches to labeled $(m+1)$ -ary trees. The region/sketch bijection is established by properties (i)–(iv) immediately before that proposition on journal pages 503–504. Moreover, the version-of-record recipe in property (iv) can create equal consecutive values: for $n=3,m=1$ , the valid sketch $\alpha_1^{(0)}\alpha_2^{(0)}\alpha_1^{(1)}\alpha_3^{(0)}\alpha_2^{(1)}\alpha_3^{(1)}$ gives $z_4=z_5=3/2$ . Bernardi's arXiv version 4 repairs the recipe by using decreasing increments $2^{-p}$ for newly initialized coordinates while retaining the version of record's already-correct nonzero branch for later levels. That repair verifies the imported bijection, but the focal citation should point to the preceding properties and disclose the corrected source form.

Bernardi, corrected arXiv version 4 ↗

Bernardi Theorem 8.8 branchApplicable and sufficient after the Section 8.1 repair

The locally maximal surjection and transitive bijection have the required scope

PDF pages 5–6 and 8–13 · Theorem 2.9 as used in Theorems 3.2, 4.1, and 5.1 · arXiv:2603.24886v1

Bernardi Lemmas 8.12–8.13 identify each arrangement region with an equivalence class of Catalan subregions and translate local maximality into the corresponding tree condition. Their proof gives surjectivity for arbitrary $S$ . When $S$ is transitive, Bernardi Theorem 4.6 supplies equal finite cardinalities, so the map is bijective. These are exactly the two assertions quoted in focal Theorem 2.9. The focal paper's use is therefore sound once the earlier region/sketch construction is taken from the corrected arXiv version.

Bernardi, Advances in Mathematics version of record ↗

Transitivity definitionIncomplete source transcription · verified repair

The focal transcription omits the source's distinct-index domain

PDF page 5 · Definition 2.7 · arXiv:2603.24886v1

Bernardi Definition 4.3 quantifies over distinct indices $a,b,c$ . Focal Definition 2.7 quantifies only $i,j$ and then uses an unbound $k$ ; without distinctness it also calls undefined diagonal sets. Quantifying pairwise distinct $i,j,k$ is both source-faithful and the domain used by the later proofs. The same explicit restriction should be propagated to Properties X and Y whenever their displayed $S^+$ terms would otherwise be diagonal.

Bernardi, Advances in Mathematics version of record ↗

Inherited counting branchSources complete for the marked branch

The transitive cardinality input terminates at two published monographs

PDF pages 5–6 and 11–13 · inherited use of Bernardi Theorem 8.8 in Theorem 2.9 and Theorem 5.1 · arXiv:2603.24886v1

Bernardi's bijectivity upgrade uses Theorem 4.6. Its proof-bearing chain passes through the signed region formula, where Zaslavsky's characteristic-polynomial evaluation converts the coboundary identity to a region count, and through Claim 7.6, where Stanley's Chapter 5.3 plane-tree/path encoding and cyclic-conjugation count evaluate the tree factor. Zaslavsky's Memoirs AMS monograph and Stanley's Enumerative Combinatorics, Volume 2 are published book terminals, so no further graph recursion is required. Contextual citations in the focal paper, including Mazin–Miller's alternative graphical proof and the earlier surjectivity result that Section 3 reproves, do not affect the marked claims.

Bernardi, exact version of record ↗

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Paper: arXiv:2603.24886v1
Authors listed: Olivier Bernardi, Neha Goregaokar
Audit date: August 23, 2026

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Bijectivity of a generalized Pak-Stanley labeling

Paper metadata

Abstract

Dependence graphs

Bijectivity of a generalized Pak-Stanley labeling

Audit summary

Detailed mathematical audit

The parking-function domain and the off-diagonal arc set must be stated

Index and empty-set conventions are missing from the main scope

The right-inverse identity survives the definition repairs

The sufficient bijectivity theorem uses only the sound half of Lemma 4.4

The transitive classification is false in the stated positive-n scope

The terminal queue coordinate is off by one

The two converse witnesses are specified only by ellipses

The injectivity induction is valid after the distinct-index convention

The positive-shift witness contains a false membership claim and unproved completions

The parameter reconstruction has local errors and cannot cover n=2

The focal pinpoint is wrong and the cited journal construction needs the later repair

The locally maximal surjection and transitive bijection have the required scope

The focal transcription omits the source's distinct-index domain

The transitive cardinality input terminates at two published monographs

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