Let $q \geq 2$. What does the expression $(q^n1)(q^nq)(q^nq^2)(q^nq^3)\ldots(q^nq^{n1})/n!$ count? If $q$ is a prime power, then this is the number of bases of an $n$dimensional vector space over a field with $q$ elements.

This is a partial answer...perhaps someone will improve on it! Historically, most of the $q$analog formulae (beginning from Euler) were derived based on the assumption that $q< 1$ (to ensure series convergence) or $q=p^k$ for a prime $p$. John Baez in one of his weekly finds (week184) discusses the geometric interpretation of $q=1$ (counting over $\mathbb CP^n$), $q=1$ (counting over $\mathbb RP^n$) and $q=$a prime power (counting over PG($\mathbb F_q$)). There is no discussion for other values of $q$. However, in Gasper and Rahman's Basic Hypergeometric Series, there is an inversion identity on page 4 which can be used when $q > 1$:
This returns a new expression in base $1/q < 1$. You can see some examples of the identity being applied in Gasper's Lecture Notes on qseries (Exercise 1.1, Exercise 2.3, pg 14) I have no idea how to interpret the result geometrically. Could it be relevant to Buildings, Buekenhout geometry or $p$adic geometry? 

