Published paper

Deformations of the braid arrangement and trees

Olivier Bernardi

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Publication metadata

Journal citation: Advances in Mathematics 335, 466–518 (2018)
Publication date: 2018-07-18
DOI: 10.1016/j.aim.2018.07.020 ↗
arXiv counterpart: arXiv:1604.06554v4

Abstract

We establish general counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form $x_i-x_j=s$ for some integer $s$ . Classical examples include the braid, Catalan, Shi, semiorder and Linial arrangements, as well as graphical arrangements. We express the number of regions of any such arrangement as a signed count of decorated plane trees. The characteristic and coboundary polynomials of these arrangements also have simple expressions in terms of these trees. We then focus on certain well-behaved deformations that we call transitive and establish, for every such deformation, a simple bijection between its regions and a set of labeled plane trees defined by local conditions.

Role in dependence graphs

Proof-critical source

The geometry of a counting formula for deformations of the braid arrangement

This paper is included only for the following marked statements:

Proposition 8.1 · journal pp. 506–507Bijection between annotated m-sketches and labeled (m+1)-ary trees, including the consecutive-letter/child characterization used again in the local-maximality argument.
Theorem 8.8 and Lemmas 8.12–8.13 · journal pp. 512–513The region-to-tree map is surjective for every tuple S; for transitive S, Theorem 4.6 supplies equinumeracy and promotes it to a bijection. This is the exact region theorem imported by Bernardi 2025.

Proof-critical source

Bijectivity of a generalized Pak-Stanley labeling

This paper is included only for the following marked statements:

Catalan-region and annotated-sketch bijection · Adv. Math. 335 (2018), pp. 503–504, properties (i)–(iv) immediately before Proposition 8.1Identifies every (m,n)-sketch with one Catalan region and makes beta_S, Phi, and the focal right-inverse construction geometrically meaningful.
Theorem 8.8 together with Lemmas 8.12–8.13 · Adv. Math. 335 (2018), pp. 511–513Gives surjectivity from locally maximal sketches to regions for arbitrary S and upgrades it to bijectivity when S is transitive.
Theorem 4.6 · Adv. Math. 335 (2018), pp. 478–480; proof-bearing chain through Theorem 5.2 and Sections 5–7, pp. 480–502Supplies the equal-cardinality step that turns the Section 8.3 surjection into a bijection for transitive arrangements.

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Version of record · Advances in Mathematics 335 (2018), 466–518

Olivier Bernardi. Deformations of the braid arrangement and trees. Advances in Mathematics 335 (2018), 466–518.

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01Statements3 reported findingsCorrect

The boxed-tree formulas for region counts and coboundary polynomials, the transitive-arrangement equinumeracy results, and the region-to-tree bijection of Theorem 8.8 are correct. The published proof of the last result contains a defective explicit inverse construction, but the repaired construction in arXiv:1604.06554v4 verifies the theorem rather than changing its statement.

Theorems 4.2, 4.6, and 5.2Correct

The boxed-tree counting and coboundary formulas are supported

Journal pages 478–502 · Theorems 4.2, 4.6, 5.2 and Sections 5–7

Central subarrangements are encoded by weighted graphs, the multivariate exponential formula isolates connected components, and the gas/tree decomposition evaluates the resulting formal series. Zaslavsky's characteristic-polynomial evaluation then converts the coboundary identity into the signed region count. For transitive tuples, the local cadet condition makes the remaining signed terms cancel in pairs, leaving exactly the claimed family of trees. The formal-series limits are coefficientwise and the finite-support estimates used before taking them are adequate.

Exact Advances in Mathematics version of record ↗

Proposition 8.1Correct

Annotated sketches are bijective with labeled plane trees

Journal pages 506–507 · Proposition 8.1 and proof

The bud-growth construction creates non-bud vertices in the tree order used by the proposed inverse, so reading the children in that order recovers every annotated sketch. Conversely, annotation reconstructs the same growth sequence. The consecutive-letter clause follows directly from whether the relevant child is a node or a leaf. The proof contains harmless local symbol slips—its input word belongs to $D^{(m)}(n)$ , and the final child test uses the exponent of the following letter—but these have unique corrections and do not affect the proposition.

Theorem 8.8Correct after verified repair

The general transitive region bijection is correct after the Section 8.1 repair

Journal pages 503–513 · property (iv), Proposition 8.1, Lemmas 8.12–8.13, and Theorem 8.8

Catalan regions are encoded by annotated sketches and then by trees. Lemma 8.12 gives a unique $S$ -maximal Catalan subregion in every $S$ -region, while Lemma 8.13 identifies $S$ -local maximality with membership in $T_S$ . Thus $\Psi_S$ is surjective for every $S$ . When $S$ is transitive, Theorem 4.6 gives equal finite cardinalities and promotes the surjection to a bijection. The only material defect is the explicit inverse recipe in property (iv); replacing its fixed increments by the decreasing increments printed in arXiv v4 makes the Catalan-region encoding bijective and completes the chain.

Corrected arXiv version 4 ↗

02Proofs3 reported findingsContains incorrect or incomplete proofs

The version of record's explicit inverse in Section 8.1 is not a valid construction: its fixed increments can make two successive ordered values equal. This affects the published proof of the Catalan-region/sketch bijection and hence the setup of Theorem 8.8. The defect has a verified repair in arXiv version 4; the rest of the counting and bijection arguments then close.

Section 8.1, property (iv)Incorrect as written · verified repair

The printed inverse can fail to produce strictly increasing coordinates

Journal pages 503–504 · property (iv) immediately before Proposition 8.1

The version of record assigns an increment $1/(n+1)$ whenever a new zero-indexed letter appears. For $n=3$ , $m=1$ , the valid annotated sketch $\alpha_1^{(0)}\,\alpha_2^{(0)}\,\alpha_1^{(1)}\,\alpha_3^{(0)}\,\alpha_2^{(1)}\,\alpha_3^{(1)}$ yields $z_1=1/4$ , $z_2=1/2$ , $z_3=5/4$ , $z_4=3/2$ , and $z_5=3/2$ , contradicting the required strict order $z_1<\cdots<z_6$ . ArXiv:1604.06554v4 gives the verified repair to that branch: with $z_0=0$ , set $z_p=z_{p-1}+2^{-p}$ for a letter $\alpha_i^{(0)}$ , while retaining the version of record's rule $z_p=x_i+s$ for $s\ne0$ . The tail of the decreasing geometric series is smaller than the next integer displacement, so all prescribed comparisons remain strict and the annotated order is realized.

Bernardi, corrected arXiv:1604.06554v4, author-copy pages 34–35 ↗

Theorem 8.8 proof chainCorrect and complete after verified repair

Surjectivity and the transitive bijectivity upgrade are complete after repair

Journal pages 512–513 · Lemmas 8.12–8.13 and proof of Theorem 8.8

An $S$ -move crosses exactly one Catalan wall absent from $A_S$ , so equivalence classes are precisely the $A_S$ -regions and each finite class has one lexicographic maximum. Proposition 8.1 translates every improving move into the forbidden cadet condition, proving Lemma 8.13. Every global maximum is local, which gives surjectivity without transitivity. For transitive $S$ , Theorem 4.6 supplies equal cardinalities; its proof-critical inputs are Zaslavsky's region-count formula and the exponential/tree-path tools cited to Stanley's Enumerative Combinatorics, Volume 2. Hence the final bijectivity step is justified.

Sections 3–7Correct and complete

The enumerative proof is internally consistent

Journal pages 472–502 · signed involutions, coboundary generating functions, and their proofs

The cadet-sequence definitions match the forbidden-intercept conditions in each orientation, the sign-reversing involutions pair exactly the non-transitive tree contributions, and the coefficient extractions from the multitype series recover the fixed-dimension arrangements. The graph and tree encodings are reversible, and the energy statistic reproduces the coboundary weight. No additional unresolved proof gap was found in the full manuscript.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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