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Original submission
March 1, 2025
Claimed MSC
arXiv categories
math.CO

Abstract

We establish a general bijective framework for encoding faces of some classical hyperplane arrangements. Precisely, we consider hyperplane arrangements in Rn\mathbb{R}^n whose hyperplanes are all of the form {xi−xj=s}\{x_i-x_j=s\} for some i,j∈[n]i,j\in[n] and s∈Zs\in\mathbb{Z}. Such an arrangement AA is strongly transitive if it satisfies the following condition: if {xi−xj=s}∉A\{x_i-x_j=s\}\notin A and {xj−xk=t}∉A\{x_j-x_k=t\}\notin A for some i,j,k∈[n]i,j,k\in[n] and s,t≥0s,t\geq0, then {xi−xk=s+t}∉A\{x_i-x_k=s+t\}\notin A. For any strongly transitive arrangement AA, we establish a bijection between the faces of AA and some set of decorated plane trees.

Dependence graphs

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Audit summary

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
01Statements2 reported findingsCorrect

The marked-tree bijection for faces of strongly transitive braid-type arrangements, its codimension statistic, the specializations to Catalan, Shi, semiorder, and intermediate arrangements, and the face-enumeration consequences are correct at the stated scope.

Theorem 3.8Correct

The face bijection has the advertised scope and dimension statistic

PDF pages 6 and 21–28 · Theorem 3.8 and proof · arXiv:2503.00662v1

Every face is represented as a region of the arrangement restricted to its affine span. Lemmas 6.1–6.2 identify that restriction with a strongly transitive braid-type arrangement, so the exact Bernardi region bijection applies on each flat. Lemmas 6.3–6.4 identify flats with marked-edge contractions, and Lemma 6.5 makes the resulting square commute. The number of marked edges is the codimension because contracting the marked forest reduces the number of blocks by exactly that number. Thus the construction is bijective and preserves the claimed statistic.

Exact arXiv version 1 ↗
Sections 4–5Correct

The classical specializations and face generating functions follow

PDF pages 8–19 · Propositions 4.1–5.5 · arXiv:2503.00662v1

The m-Catalan, m-Shi, semiorder, and interpolating arrangements satisfy the displayed strong-transitivity tests, and Theorem 3.8 specializes to the stated decorated-tree families. In the symmetric cases, marking eligible edges gives the factor uu per marked edge; the recursive root decomposition therefore produces the displayed face-enumeration equations. The Catalan specialization agrees with Levear's independent bijection and generating function.

02Proofs3 reported findingsCorrect

The restriction-to-flats proof, its marked-tree reconstruction, and the generating-function arguments are complete. A single displayed sign in Lemma 6.1 is wrong, but the definition immediately above and the coordinate substitution give a unique verified correction, and the proof thereafter uses the corrected expression.

Proof of Theorem 3.8Correct and complete

The restriction and marked-tree bijections form a valid commutative diagram

PDF pages 21–28 · Section 6 · arXiv:2503.00662v1

The proof partitions coordinates into the blocks of a flat, transports the restricted arrangement to one coordinate per block, proves preservation of strong transitivity, and pairs the same block data with marked connected components of the tree. The two constructions are inverses on each stratum and their compatibility with the region map is checked hyperplane by hyperplane. This proves both injectivity and surjectivity rather than relying only on enumeration.

Exact arXiv version 1 ↗
Lemma 6.1Typo · verified repair

One sign in the displayed image hyperplane is reversed

PDF page 20 · proof of Lemma 6.1 · arXiv:2503.00662v1

The proof prints the image of {xi−xj=s}\{x_i-x_j=s\} as {xk−xℓ=s−δL(i)−δL(j)}\{x_k-x_\ell=s-\delta_L(i)-\delta_L(j)\}. Since xi=xak+δL(i)x_i=x_{a_k}+\delta_L(i) and xj=xaℓ+δL(j)x_j=x_{a_\ell}+\delta_L(j), the correct equation is xk−xℓ=s−δL(i)+δL(j).x_k-x_\ell=s-\delta_L(i)+\delta_L(j). This is exactly the definition of A~L\widetilde A_L displayed immediately before the lemma and the later argument uses that definition. Replacing the second minus by a plus repairs the line without changing any result.

Use of Bernardi 2018Correct after verified source repair

The imported region bijection is valid after its published-source repair

PDF pages 3 and 21–22 · Theorem 2.9 and proof of Theorem 3.8 · arXiv:2503.00662v1

The exact 2018 version of record proves surjectivity and uses its region-count theorem for injectivity. Its Section 8.1 inverse recipe has a fixed-step defect, but arXiv:1604.06554v4 replaces the problematic increment by 2−p2^{-p} and corrects the associated zero-index condition. The corrected source establishes the exact theorem used here, so the downstream proof is sound.

Bernardi arXiv version 4 with the Section 8.1 correction ↗
03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2503.00662v1
Authors listed
Olivier Bernardi
Audit date
August 23, 2026
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