OEIS A000130: Exactly one rising or falling adjacency in permutations

A deep dive into A000130, the sequence that equals half the number of length-n permutations with exactly one rising or falling succession. We walk through a concrete n=3 example, clarify what rising and falling successions are, and explain how the sequence grows. We discuss the handy factorial-based approximation e^{-2} n!, and why that hints at deeper structure in these permutations. We also connect A000130 to pattern avoidance in permutations and its role in combinatorics and number theory, showing how a seemingly abstract rule reveals rich mathematical relationships.

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