  Motivated by recent work on mixtures of classical and free probabilities, we
introduce and study the notion of $\epsilon$-noncrossing partitions. It is
shown that the set of such partitions forms a lattice, which interpolates as a
poset between the poset of partitions and the one of noncrossing partitions.
Moreover, $\epsilon$-cumulants are introduced and shown to characterize the
notion of $\epsilon$-independence.
