Counting with Symmetries

Suppose you are walking through a carnival when a booth attendee named Pauly calls you over to a tent with a challenge: Pauly hands you some string and red and green beads, and tells you to make exactly one of every different bracelet that you can with exactly 6 beads on each. After a few minutes, you hand Pauly your bracelets, three of which are pictured below—but when you hand in your work, Pauly begins saying that the left two of your “different” bracelets were actually the same! Is there a way for you and Pauly to both be correct and reconcile your differences?

Realizing “different” is different for you than for Pauly, it is possible to reconcile. In our project, we will study the content of Pólya’s Enumeration Theorem, which studies problems in Combinatorics where there are symmetries involved. In the case of Pauly’s bracelets, you were sure your bracelets were different because you assumed only rotational symmetries (any two bracelets that can be rotated to each other are counted as the same), but Pauly had assumed more symmetries which included flipping the bracelets over. That’s why you thought the two left bracelets were different, but Pauly disagreed!

Beyond these bracelets, Pólya’s Enumeration Theorem has applications almost any- where one would study the symmetries of a finite number of things. Our goal is to understand the theorem, particularly understand how to apply it, and then come up with new applications in different fields. These might include applications in count- ing problems that one can find in the real world, applications within mathematics, or applications to other fields including chemistry. Previous experience with permutations is recommended, and experience with combinatorics/counting is a plus but not a requirement!

Are the top two bracelets different?