Published paper
Abstract
This published chapter develops the intersection-poset and characteristic-polynomial theory of hyperplane arrangements. Proposition 2.5 gives the graphical-arrangement region and acyclic-orientation bijection, and Theorem 2.7 identifies the graphical and chromatic polynomials.
Role in dependence graphs
Proof-critical source
Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements
This paper is included only for the following marked statements:
- Proposition 2.5 · printed p. 418; chapter PDF p. 30Bijection between regions of a graphical arrangement and acyclic orientations.
- Theorem 2.7 · printed pp. 414–418; chapter PDF pp. 26–30Identifies the graph chromatic polynomial with the graphical-arrangement characteristic polynomial.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact published chapter · IAS/Park City Mathematics Series 13 (2007), 389–496
Richard P. Stanley. An Introduction to Hyperplane Arrangements. In Geometric Combinatorics, IAS/Park City Mathematics Series 13 (2007), 389–496.
Open audited source ↗01Statements3 reported findingsCorrect
The chapter's intersection-poset, characteristic-polynomial, region-count, graphical-arrangement, finite-field, freeness, and braid-deformation results are correct within their stated scope. No false theorem or proposition was found in the complete published chapter.
Theorems 2.4–2.7Correct
Characteristic and chromatic polynomial identities are correct
Printed pages 413–419 · Theorems 2.4–2.7 and Proposition 2.5
Whitney's subset expansion and Zaslavsky's evaluations have the correct signs for the chapter's intersection-poset convention. Orienting each graph edge toward the larger coordinate is constant on an arrangement region and produces exactly the acyclic orientations; a topological ordering constructs the inverse region. Proper colorings then give the graphical-arrangement characteristic polynomial. These are the exact focal inputs.
Exact published chapter ↗Lectures 1–3Correct
The main arrangement invariants and examples are internally consistent
Printed pages 389–453 · Lectures 1–3
Deletion–restriction, finite-field counting, broken-circuit coefficients, and the braid and Shi arrangement examples all use compatible rank and essentialization conventions. The examples reproduce the known factorizations and region counts, including their nonessential factors.
Lectures 4–5Correct
The freeness and deformation statements preserve their hypotheses
Printed pages 454–496 · Lectures 4–5
The freeness discussion distinguishes sufficient factorization results from converses, and the braid-deformation formulas are labeled at the correct level of theorem, example, or exercise. No conjectural assertion is presented as proved.
02Proofs3 reported findingsCorrect
Proofs and proof sketches throughout the chapter were checked at the level claimed by this published lecture text. The two marked graphical-arrangement results are fully proved, and every later use relevant to this dependency chain is supported. No unresolved proof gap was found.
Proposition 2.5Correct
The region and acyclic-orientation maps are inverse
Printed page 418 · Proposition 2.5 and proof
Coordinates in a region determine a strict direction on each edge. A directed cycle would force an impossible strict inequality cycle. Conversely an acyclic orientation has a linear extension, and assigning increasing coordinates in that order produces a point in a unique region inducing the orientation. Convexity shows the construction is independent of the chosen point.
Theorem 2.7Correct
Finite-field coloring gives the polynomial equality
Printed pages 414–418 · finite-field method and Theorem 2.7
Over every sufficiently large finite field, the complement of the graphical hyperplanes consists exactly of proper vertex colorings. Both counts are polynomials in the field size, so equality on infinitely many prime powers proves equality of the chromatic and arrangement characteristic polynomials.
Deletion–restriction frameworkCorrect
Inductive formulas treat loops and rank changes correctly
Printed pages 397–415 · intersection posets, deletion–restriction, and characteristic polynomial
The recurrence is stated only when deletion and restriction are arrangements of the claimed ranks, with separate handling of redundant hyperplanes. Möbius inversion supplies the same recurrence, so the later region and coloring arguments rest on a complete induction.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.