OEIS A000131: Asymmetrical triangulations of a polygon

In this episode we explore A000131, which counts asymmetrical (chiral) triangulations of a convex n-gon. Starting at n=7 with two examples (the heptagon) and n=8 with five, the numbers grow rapidly as the number of sides increases. The counting uses Catalan numbers and an inclusion–exclusion sieve to remove triangulations that exhibit symmetry, leaving only the asymmetrical ones. We’ll trace how Catalan numbers enter the recursive structure, discuss Richard Guy’s foundational work, and touch on the broader family of related sequences that arise when we relax symmetry constraints.

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