OEIS A000240: One-Fixed-Point Permutations

Delving into A000240—the number of permutations of n items with exactly one fixed point (a Rencontres number). We’ll trace the journey from derangements (no fixed points) to these partial fixed-point counts, show the simple relation dn,1 = C(n,1) · D(n−1), and explore why, as n grows, the probability of no fixed points and the probability of exactly one fixed point both tend to 1/e. Along the way we’ll connect counting, probability, and the surprising unity of randomness and order in permutations, drawing on MathWorld, OEIS, Wikipedia, Missouri State, and Reddit.

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