Proof-critical dependence graph
The geometry of a counting formula for deformations of the braid arrangement
Statement-restricted proof-dependence graph for the contribution-one theorem and its tree-contribution consequences. The graph follows the exact face-bijection input through Bernardi's region bijection and its counting step. It excludes background, historical citations, independently sufficient alternative sources, and results reproved inside the focal paper.
Graph scope5 nodes5 proof-critical linksChecked August 23, 2026
Oriented proof graph
Dependence map
Arrows point from a prerequisite toward the paper whose marked statement uses it.
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Every visible arrow is documented in the evidence ledger below. A non-book leaf means that no earlier proof-critical source was identified for the marked statement—not that the paper has no other citations.
Evidence ledger
Proof-critical links
Bijections for faces of braid-type arrangements→The geometry of a counting formula for deformations of the braid arrangementHeadline lineageVerified
The focal restriction map starts from the exact marked-tree/face bijection for the m-Catalan arrangement. That bijection supplies both bijectivity and the marked-edge/codimension relation used to identify the sign with face dimension.
Citation location: Focal Theorem 3.12 and proof of Lemma 3.17, PDF pp. 12–15, citation [1, Theorem 3.8]; Bernardi 2025 Theorem 3.8, PDF p. 6, proof pp. 21–28.- Theorem 3.1Shows that every region of an arbitrary deformation of the braid arrangement contributes 1 to Bernardi's signed tree count.
- Lemma 3.17Restricts the m-Catalan face bijection to S-boxed trees and faces not contained in a hyperplane of the deformation, preserving dimension as the number of boxes.
Deformations of the braid arrangement and trees→Bijections for faces of braid-type arrangementsHeadline lineageVerified
Bernardi 2025 views every face as a region of a restricted arrangement. Lemmas 6.1–6.2 identify each restriction as a strongly transitive braid-type arrangement, and the cited Bernardi 2018 region bijection then gives the needed tree-to-region bijection on every flat.
Citation location: Bernardi 2025 proof of Theorem 3.8, PDF pp. 21–22, citing Theorem 2.9 [Ber18]; Bernardi 2018 Theorem 8.8, journal pp. 512–513.Verification note: The exact 2018 version of record and its corrected arXiv v4 were compared.- Theorem 3.8Bijection between marked trees and faces of every strongly transitive braid-type arrangement, with marked edges recording codimension. Its m-Catalan specialization is the focal Theorem 3.12.
Deformations of the braid arrangement and trees→The geometry of a counting formula for deformations of the braid arrangementHeadline lineageVerified
The focal paper directly imports Bernardi's tree-to-m-Catalan-region bijection as Theorem 2.9 to define the tree set attached to an -region. In Lemma 3.17 it uses incidence of the marked face with that unmarked Catalan region to restrict the global face bijection region by region.
Citation location: Focal Theorem 2.9 and Definition 2.10, PDF p. 6; Remark 3.14 and final region-restriction step in Lemma 3.17, PDF pp. 13 and 15; Bernardi 2018 Theorem 8.8, journal pp. 512–513.Verification note: The exact version of record was checked against the repaired arXiv v4.- Theorem 3.1Shows that every region of an arbitrary deformation of the braid arrangement contributes 1 to Bernardi's signed tree count.
- Lemma 3.17Restricts the m-Catalan face bijection to S-boxed trees and faces not contained in a hyperplane of the deformation, preserving dimension as the number of boxes.
Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes→Deformations of the braid arrangement and treesHeadline lineageTerminal source
Theorem 8.8 obtains surjectivity internally and uses Theorem 4.6 for equal cardinalities. Theorem 4.6 uses the boxed-tree region count of Theorem 4.2; that count is extracted from Theorem 5.2 only after Zaslavsky's characteristic-polynomial evaluation converts the coboundary polynomial to the number of regions.
Citation location: Bernardi 2018 Theorem 8.8 proof, journal p. 513; Theorem 4.6 proof, p. 479; equation (5.1) and the derivation of (5.8), pp. 480–482, citation [46].Verification note: The source is a published monograph, where recursive expansion stops.- Theorem 8.8 and Lemmas 8.12–8.13The 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.
Enumerative Combinatorics, Volume 2→Deformations of the braid arrangement and treesHeadline lineageTerminal source
The equal-cardinality half of Theorem 8.8 passes through Theorems 4.6, 4.2, and 5.2. Bernardi's proof of Theorem 5.2 invokes the multivariate exponential formula in Lemma 7.3 and the standard plane-tree/path correspondence in Claim 7.6; both are cited to Stanley's book.
Citation location: Bernardi 2018 Lemma 7.3 proof, journal p. 496, citation [43]; Claim 7.6 proof, p. 498, citation [43, Chapter 5.3]; completion of Theorem 5.2, pp. 501–502.Verification note: The source is a published book, where recursive expansion stops.- Theorem 8.8 and Lemmas 8.12–8.13The 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.