We survey the surprising connections between the structure of the group, and properties of this sequence. We show that the cogrowth sequence is not P-recursive when $G$ is an amenable group of superpolynomial growth and compute the exponential growth of the cogrowth sequence for certain infinite families of free products of finite groups and free groups. Work in collaboration with Jason Bell.
In this talk we meld these two directions and discover when the chromatic symmetric function of trees is not a positive linear combination of elementary symmetric functions, using a beautiful but overlooked result of Wolfgang III from 1997.
This is joint work with Samantha Dahlberg and Adrian She.