By Shalosh B. Ekhad and Doron Zeilberger
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org)
Written: Jan. 19, 2020.
In a delightful article that recently appeared in the American Mathematics Monthly, Norbert Henze and Günter Last discuss the "Absent-Minded Passengers" Problem, but left open finding an explicit expression for the probability generating function, of the random variable "Number of passengers occupying a wrong seat", when the number of absent-minded passengers is larger than one. This is accomplished in this note, using experimental mathematics and symbolic computation. We also derive explicit expressions for the first 8 moments of the original case of one absent-minded passenger, and indicate how to extend it to the general case.
Added Jan. 22, 2020: Watch Norbert Henze's crystal-clear clip (in German) why the probability of the last passenger (in the original case) sitting where he is supposed to is 1/2, using a "mirror" gedanken experiment.
Added March 5, 2020: Carsten Schnedier met the challenge (and far beyond!). See his brilliant article. A donation of $100 to the OEIS Foundation has been made.