A partition of $[n]$ is indecomposable if no subset of its blocks partitions $[k]$ with $k \in [n-1]$. Irreducible set partitions are defined at http://oeis.org/A055105 . Both are counted by http://oeis.org/A074664 . Is any bijection between them known?
Mike Zabrocki has kindly addressed me to the following note http://www.billchen.org/publications/2011_P7_Unsplitable/2011_P7_Unsplitable.pdf where the bijection is built (here "atomic"="indecomposable" and "unsplittable"="irreducible"). Note that the correct version of the definition of splittable or reducible partition (page 3 on top) allows different lengths for the reducing partitions (a small but important detail that will be fixed in the OEIS link).