  We investigate the mixing time of the capacity $k$ simple exclusion process
(also called the partial exclusion process) of Schultz and Sandow with $m$
particles on a segment of length $N$. We show that the $k$-SEP exhibits cutoff
at time $\frac{1}{2k\pi^2}N^2\log m$. We also introduce a related complete
multi-species process that we call the $S_{k,N}$ shuffle and show that this
process exhibits cutoff at time $\frac{1}{2k\pi^2}N^2\log (kN)$. This extends
the celebrated result of Lacoin which determined the mixing time of the
symmetric simple exclusion process on a segment of length $N$ and the adjacent
transposition shuffle, and proved cutoff in both.
