OEIS A000085: Involutions and the many faces of self-inverse permutations

We explore A000085, the involution numbers that count self-inverse permutations on n elements. From the classic recursive rule a(n) = a(n-1) + (n-1)a(n-2) to a direct closed form, we’ll see how these numbers pop up in surprising corners of math. Along the way we’ll connect involutions to pairings and matchings in graphs, touch on their links to representation theory and Young tableaux, and discuss the nickname “telephone numbers” and what it hints about real-world networks. We’ll also peek at transforms like the Hankel and binomial transforms that reveal deeper connections to factorials and Pascal’s triangle. A guided tour through a deceptively simple sequence that keeps revealing new patterns and links.

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