Georgia Institute of Technology College of Sciences

There is a natural notion of `degree’ for functions from the symmetric group to the complex numbers, which translates roughly to saying the function counts certain weighted patterns. Low degree class functions have a classical interpretation in terms of the cycle structure of permutations. I will explain how to translate between pattern counts to cycle structure using a novel symmetric function identity analogous to the Murnaghan-Nakayama identity. This relationship allows one to lift many probabilistic properties of permutation statistics to certain non-uniform distributions, and I will present some results in this direction. This is joint work with Brendon Rhoades.