Example of something easier to count with $q$-analog? Are there any known examples of combinatorial objects that become easier to count by considering some kind of $q$-analog? It seems to me that it might be impossible for the problem of computing the $q$-analog directly to be (strictly) easier than enumerating the objects themselves, as we should be able to just replace $q$ everywhere with 1. However, I'd also be interested in any enumerative problems in which $q$-analogs give us some additional insight.
 A: Sometimes the answer to a combinatorial problem is given by a ratio where both numerator and denominator can be computed as polynomials in $q$ but become zero for $q=1$.
One non-trivial example: the q-number of plane partitions that fit into $a\times b\times c$ box is given by the principal specialization of a Schur polynomial — i.e. it's a ratio of two Vandermonde determinants. This yields MacMahon formula.
(Slightly more generally, it's easier to proof q-hook-content formula first and get ordinary hook-content formula by plugging $q=1$.)
A: 
It seems to me that it might be impossible for the problem of computing the $q$-analog directly to be (strictly) easier than enumerating the objects themselves, as we should be able to just replace $q$ everywhere with $1$.

It's not impossible, because the $q$ proof does not necessarily specialize to a proof at $q=1$.  Maybe some objects in the proof collapse, maybe some arguments that don't only involve counting become incorrect, and for answers that are more complicated than  setting $q=1$ in a polynomial, showing convergence and correctness of the answer as $q \to 1$ (or however else the solution at $1$ is obtained) can be more difficult than the $q$ argument.
For example, the $q$-gamma function is easier to define and handle than $\Gamma(x)$, and does not need any regularization, when $|q| \neq 1$, but justifying its relation to the gamma function and properties thereof, is more complicated.
