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The two exceptional vertices

Abstract

Two exceptional partitions complete the two-block enumeration.

This is source Proposition 3.3(iii), including the two omitted two-block partitions.

Theorem 1.1 (The complete two-block classification).

Lean statement: D5/S3/Combinatorics/Geometry/CrownOrderPolytopeTwoExceptions.crownOddBlockMerge_two_card

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Geometry/CrownOrderPolytopeTwoExceptions.crownOddBlockMerge_two_card (✓ std3). ∎

Citation. Teemu Lundström and Leonardo Saud Maia Leite (2025). Order polytopes of crown posets. DOI: 10.48550/arXiv.2504.05123. URL: https://arxiv.org/abs/2504.05123v3.

Commentary.

For n at least two, odd-block merging together with the two explicitly constructed exceptional partitions is a bijection onto all connected compatible partitions with two quotient blocks. Their cardinality is the selected-block count plus two, which supplies the exceptional geometric vertices.

References