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A mathematical notation for juggling called Siteswap has been used for decades to teach and learn juggling. Many of the mathematical properties of this notation have been studied. However, there is an alternative method to describe juggling patterns called Stack Notation. These notations convey different information about the same juggling patterns. By analyzing juggling patterns using two methods based on the different notations, we are able to relate any juggling pattern to two distinct permutations of two Symmetric groups: Sn which is the set of permutations of juggling throws, and Sb which is the set of permutations of juggling props. From here, we can discern if the set of juggling patterns of a fixed length n and fixed number of props b is surjective to Sn or to Sb. This can tell us what types of juggling patterns are possible, and under what conditions they are possible.
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Annett, Kieran, "Juggling: Siteswap and the Symmetric Group" (2021). Senior Projects Fall 2021. 29.
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