OEIS A000166: Derangements (Subfactorial Numbers)

A deep dive into derangements: the subfactorial counts of permutations with no fixed points. We traverse recursive and inclusion-exclusion formulas, the classic hat-check problem, and the surprising appearance of the base e in asymptotics. We’ll explore connections to Bell numbers, unordered necklaces, and game theory Nash equilibria, plus links to other OEIS sequences like A000255. Along the way we touch on cryptography and computer science applications, and discuss open questions such as Zihui Sun’s conjecture on when subfactorials are perfect powers. Featuring a Fields Medal–winning guest, this episode showcases how a simple counting problem opens a rich web of mathematics.

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