OEIS A000313: Three consecutive ascending pairs in permutations

In this Deep Dive, we explore A000313 from the OEIS—the count of permutations of length n with exactly three consecutive ascending adjacent pairs. We discuss why the sequence starts with zeros and first yields a nonzero term at n = 4, and how this niche counting problem unfolds into a rich toolkit: a recurrence, an explicit formula involving e and factorials, and an exponential generating function. We’ll also look at connections to other combinatorial objects (like A000166 for derangements) and structural placements (A010027 as a diagonal in a number triangle), plus a bit of historical context. It’s a showcase of how precise counts open up elegant math and broad links across the OEIS.)

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