(q,t) analogues and the general linear group

Some of the most beloved combinatorial numbers count objects associated with the symmetric group S_n, and have a q-analogue counting objects associated with the finite general linear group GL_n(F_q). On the other hand, this q-analogue often has a second interpretation as a Hilbert series arising in the invariant theory of S_n, without mentioning GL_n(F_q). We will show several interesting example of a new "(q,t)-analogue", capturing both of these interpretations. It is a Hilbert series relating to the invariant theory of GL_n(F_q), specializing when t = 1 to the q-counting interpretation for GL_n(F_q) and when q = 1 to the Hilbert series interpretation for S_n.