The Absent-Minded Passengers Problem via Computer Algebra

By Shalosh B. Ekhad and Doron Zeilberger

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(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.

## Sample Input and Output Files for AMP.txt

• If you want to see linear recurrences with polynomial coefficients for the probability generating functions with n passengers, according to the random variable "number of passengers sitting at the wrong seat" the first k of whom are absent-minded, for k=1,2,3,4
The input gives you the output.

• If you want to explicit expression for the expecation, variance, and the scaled moments up to the 8th, of the random variable "number of passengers sitting at the wrong seat" with n passengers, where only the first passenger is absent-minded
The input gives you the output.

• If you want to compare simulations to the exact answers
The input gives you the output.

• If you want to see an empirical verification of Theorem 3 in the paper
The input gives you the output.

• If you want to see an empirical verification of Theorem 4 in the paper
The input gives you the output.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger