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 $w : s_n \times \cdots  \times s_n \rightarrow s_n$.