Proof-critical dependence graph
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.
Graph scope4 nodes3 proof-critical linksChecked August 23, 2026
Oriented proof graph
Dependence map
Arrows point from a prerequisite toward the paper whose marked statement uses it.
- Solid arrow: headline proof lineage
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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
Deformations of the braid arrangement and trees→Bijectivity of a generalized Pak-Stanley labelingHeadline lineageVerified
The focal construction uses Bernardi's sketch/Catalan-region correspondence to turn the word Psi(p) into an arrangement region. The focal bijectivity arguments also use Bernardi's surjection from locally maximal sketches to regions and, in the transitive necessity theorem, its bijectivity upgrade.
Citation location: Focal Lemma 2.6, PDF p. 5, citation [2, Proposition 8.1] (imprecise); focal Theorem 2.9, PDF pp. 5–6, citation [2, Section 8]. Exact operative VOR locators are pp. 503–504 and 511–513.Verification note: The exact version-of-record audit is published on the linked journal-paper page.- Theorem 3.2 and the right inverse beta_S composed with PsiConstructs a parking-function-to-region right inverse for the generalized Pak-Stanley labeling and gives the focal proof of surjectivity.
- Theorem 4.1Shows that every (m, epsilon)-arrangement has a bijective generalized Pak-Stanley labeling with the explicit inverse.
- Theorem 5.1Characterizes, among transitive deformations, exactly those arrangements for which the generalized Pak-Stanley labeling is bijective.
Facing up to arrangements: face-count formulas for partitions of space by hyperplanes→Deformations of the braid arrangement and treesHeadline lineageTerminal source
Bernardi uses Zaslavsky's characteristic-polynomial evaluation to obtain the region generating function (5.8). Theorem 4.2 is its fixed-dimension coefficient, and Theorem 4.6 applies the transitive cancellation to that count.
Citation location: Bernardi VOR p. 480, equation (5.1), citation [46]; Theorem 5.2 and the extraction described on p. 482; Theorem 4.6 proof pp. 478–480.- Theorem 4.6Supplies the equal-cardinality step that turns the Section 8.3 surjection into a bijection for transitive arrangements.
Enumerative Combinatorics, Volume 2→Deformations of the braid arrangement and treesHeadline lineageTerminal source
Bernardi's Claim 7.6 uses the plane-tree/Lukasiewicz-path bijection and its cyclic-conjugation count to enumerate tuples of plane trees. That claim feeds Lemma 7.5, Theorem 5.2, the signed region formula, and finally Theorem 4.6.
Citation location: Bernardi VOR pp. 497–498, Claim 7.6, citation [43, Chapter 5.3] and the immediately following cycle-lemma invocation; downstream use pp. 498–502.- Theorem 4.6Supplies the equal-cardinality step that turns the Section 8.3 surjection into a bijection for transitive arrangements.