OEIS A000303: Permutations with longest increasing run of length 2

We dive into A000303, the counting sequence for permutations of {1,...,n} whose longest increasing run (a maximal consecutive block that is strictly increasing) has length exactly 2. We unpack what an increasing run is, why runs of length 3 or more are forbidden while at least one run of length 2 must occur, and how the first terms 0, 1, 4, 16, 69 arise. We'll discuss how such counts are derived—via recursive relations, generating functions, and sometimes computer enumeration—and why this small constraint leads to rich combinatorial structure. We'll place A000303 in the broader OEIS web, note historical context (early work by Kendall, Sloan, and later refinements), and mention avenues for ongoing research, such as generalizations to runs of length 3 or other forbidden patterns, and connections to algorithmic analysis. If you enjoy precise definitions shaping surprising patterns, this is a perfect example.

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