Written: Oct. 20, 2024

In honor of our two heroes, Don Knuth who was 85, a year ago, and Neil Sloane, who is now

One of the most important sequences in enumerative combinatorics is OEIS sequence A85, the number of involutions of length n. In the Art of Computer Programming, vol. 3, Don Knuth derived the O(1/n) asymptotic formula for these numbers. In this modest tribute to our two heroes, Neil Sloane who just turned 85, and Don Knuth who was 85 a year ago, we go all the way to an O(1/n^85) asympotic formula.

This paper is accompanied by a Maple package

A85.txt that

enables the user to find the O(1/n^(k+1)) asymptotics for any 1 ≤ k ≤ 169.

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