This engaging article takes the reader on a journey in which a graphical representation of permutations provokes an interesting counting problem with surprising results. The article begins with a clear definition and visualiza- tion of permutations. A well-chosen permutation of the digits 1-9 illustrates patterns within permutations, prompting the question of how many permu- tations avoid a given pattern. The article continues with progressively more complicated examples which prepare the reader for two surprising examples and an unsolved problem.

There are unexpected connections made to famous landmarks as the journey unfolds. A computer science problem from Donald Knuth motivates the counting question. A nicely illustrated explanation of a recursive solution of one counting problem leads directly to the Catalan numbers. A related problem is solved with a familiar recurrence relation, the generator of the Fibonacci numbers. The article concludes with some intriguing clues to send the reader on a journey into more complicated and unsolved problems.

This video was recorded at the 2023 MAA MAthFest conference where the speaker recived the Trevor Evans Award, established by the Board of Governors in 1992 and first awarded in 1996. It is made to authors of expository articles accessible to undergraduates and published in Math Horizons. The Award is named for Trevor Evans, a distinguished mathematician, teacher, and writer at Emory University.

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