I describe an empirical-yet-rigorous, algorithm, based on Riordan's rook
polynomials and the so-called C-finite ansatz, fully implemented in the
accompanying Maple package
(http://www.math.rutgers.edu/~zeilberg/tokhniot/MENAGES ), MENAGES, that
reproduces in a few seconds, rigorously-proved enumeration theorems on
permutations with restricted positions, previously proved by quite a few
illustrious human mathematicians, and that can go far beyond any human
attempts.Comment: 15 pages. An extended version of the last of three invited talks
given by the author at the 71th Seminaire Lotharingien de Combinatoire, that
took place in Bertinoro, Italy, Sept. 16-18, 201
