OEIS A000007: Symmetry in Polygon Triangulations

We explore A000007, counting inequivalent ways to dissect an (n+2)-gon into n noncrossing triangles (rotations and reflections identified). We show how this equals the number of unlabeled maximal outer planar graphs on n+2 vertices, and how the same count surfaces in hexaflexagons, associahedra, and even certain hyperbolic tilings. Along the way we glimpse Burnside's lemma in action and the unity of geometry, graph theory, and combinatorics behind a surprisingly simple counting problem.

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