Published paper
Non-crossing partitions for classical reflection groups
MSC 2020
06A0706A08
Abstract
The paper introduces the type and type analogues of non-crossing partitions. Its opening type- construction identifies the intersection lattice of the type- Coxeter arrangement with sign-stable set partitions having at most one zero block.
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 statement:
- Type-B partition-lattice identification · journal pp. 197–199Identifies intersections of type-B reflection hyperplanes with sign-stable partitions and a possible zero block.
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Audit summary
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Exact reviewed source
Exact version of record · Discrete Mathematics 177 (1997), 195–222
Victor Reiner. Non-crossing partitions for classical reflection groups. Discrete Mathematics 177 (1997), 195–222.
Open audited source ↗01Statements3 reported findingsCorrect
The type- and type- noncrossing-partition models, lattice identifications, rank enumerations, Möbius values, and classical fixed-point interpretations are correct in the complete version of record. No false mathematical statement was found.
Type-$B$ intersection latticeCorrect
Sign-stable partitions correctly encode type- flats
Journal pages 197–199 · opening construction and type- partition definition
An intersection of hyperplanes , , and identifies signed indices in paired blocks, with at most one block fixed by negation. Conversely every such partition defines exactly the corresponding coordinate-equality flat. Refinement reverses flat inclusion as required. This is the exact fact used by the focal type- face discussion.
Exact version of record ↗Theorems 2–4Correct
The rank numbers and Möbius invariants agree with the lattice models
Journal pages 201–215 · type- and type- enumeration theorems
The parenthesization and annular models give reversible encodings of noncrossing partitions. Counting block choices yields the stated Narayana-type rank numbers, and summing the rank-selected chains gives the displayed zeta-polynomial and Möbius evaluations. Small-rank boundary cases are separated where the type- model degenerates.
Section 5Correct
The classical-group comparison is consistent with the combinatorial constructions
Journal pages 215–221 · reflection-group and fixed-point discussion
The interval below a Coxeter element in absolute order has the same rank and covering relations as the constructed noncrossing lattice in the classical types. The exceptional small cases are not silently folded into the generic formulas.
02Proofs2 reported findingsCorrect
The manuscript's full proof chain was checked. The diagram models are invertible, their crossing criteria match lattice refinement, and the enumerations follow from explicit finite choices or standard generating identities. No proof-critical gap was found.
Type-$B$ bijectionsCorrect
The circular model preserves blocks, zero blocks, and noncrossingness
Journal pages 199–208 · type- constructions and proofs
Placing signed elements antipodally makes negation-stable blocks visible as either antipodal pairs or one central zero block. Convex hulls cross exactly when the corresponding partition violates the stated order condition. Erasing signs and reconstructing antipodes are mutually inverse on the admissible class.
Type-$D$ annular argumentCorrect
The treatment of the central signed pair covers the exceptional cases
Journal pages 208–216 · type- construction and enumeration
Moving the final signed pair to the inner boundary distinguishes the two configurations that collapse in a single circle. The proof tracks whether those inner vertices lie in a zero block or in paired blocks, so the inverse construction is well-defined and the enumeration does not double count.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.