Generating all equal-sized set partitions
Suppose $N=\{0,\dots,N-1\}$.
There are $\binom{N-1}{K-1}$ choices for which elements are equivalent to 0. For each choice of these there are $\binom{N-K-1}{K-1}$ choices for which elements are equivalent to the smallest number not equivalent to 0. And then $\binom{N-2K-1}{K-1}$ for which elements are equivalent to the smallest number not equivalent to any number chosen so far. 
Continuing to iterate this way, we get through all the $\prod_{a=0}^{N/K-1}\binom{N-aK-1}{K-1}$ balanced partitions.