Schur-Weyl duality involves the commuting actions of the general linear group and the symmetric group on a tensor space, relating the irreducible representations of these two groups. The idea can be generalised to other groups using the partition algebra and its subalgebras. I will discuss one such generalisation, `Schur-Weyl-Jones duality', as well as a refinement of this used to obtain a combinatorial formula for irreducible characters of the symmetric group. Time permitting, I will discuss an application of this formula towards obtaining new bounds on the expected irreducible character of a wrandom permutation, that is, a random permutation obtained via a word map $w : S_n \times \cdots  \times S_n \rightarrow S_n$.