OEIS A00239: Permutations with one run

We use the tiny, two-term sequence of one-run permutations as a doorway into analytic combinatorics. This episode sketches how generating functions (both exponential and ordinary) and the symbolic method turn simple structural questions into powerful algebra, with permutation cycles at the heart of the story. We explain why permutations are naturally a set of cycles, how labeled structures lead to exponential generating functions, and how singularity analysis yields growth rates for large n. Beyond the one-run case, we glimpse how the same toolkit counts broader run structures, connects to valleys and atomic components, and why these ideas matter in algorithms, physics, and biology.

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