The Aldous order is an ordering of representations of the symmetric group motivated by the Aldous Conjecture, a conjecture about random processes proved in 2009. In general, the Aldous order is very difficult to compute, and the proper relations have yet to be determined even for small cases. However, by restricting the problem down to Young-Jucys-Murphy elements, the problem becomes explicitly combinatorial. This approach has led to many novel insights, whose proofs are simple and elegant. However, there remain many open questions related to the Aldous Order, both in general and for the Young-Jucys-Murphy elements.

Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe how to use permutation representations of the symmetric group to create and study efficient algorithms that yield such decompositions.