OEIS A000274: Two consecutive ascending pairs and exceedances in derangements

We explore the OEIS sequence A000274, the count of permutations of length N with two consecutive ascents (two adjacent ascending pairs), and Deutsch’s alternative definition as the total number of exceedances across all derangements of {1,…,N}. We clarify what a derangement is and what counts as an exceedance, illustrate why N=3 yields 1, and outline how to generate terms without listing every permutation: a four-term recurrence, an exponential generating function, and a conjectured formula involving e. We also note the neat connection to derangements via a Mathematica-style expression, place A000274 in the broader OEIS network of related sequences, and touch on its historical roots in classic combinatorics. A final takeaway: two different counting problems arrive at the same sequence, hinting at deeper structure in combinatorics.

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