Is there a bijection from indecomposable to irreducible set partitions?

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?

• If I understand these correctly, indecomposable is, for each $k$, there exists a block which contains numbers both $\geq k$ and $\leq k$. Irreducible is, for each $k$, there exists a block that does not contain numbers both $\geq k$ and $\leq k$. And they are equinumerous? Interesting! – Will Sawin May 8 '14 at 17:48
• For $n=3$, I see four irreducible partitions out of five, namely $\big\{\{1\},\{2\},\{3\}\big\}$, $\big\{\{1,2\},\{3\}\big\}$, $\big\{\{1,3\},\{2\}\big\}$, $\big\{\{1\},\{2,3\}\big\}$. But I only see two indecomposable partitions: $\big\{\{1\, 3\},\{2\}\big\}$, $\big\{\{1,2,3\}\big\}$. What am I missing? – Pietro Majer May 8 '14 at 18:53
• @Will: I guess you mean $>$ instead of $\ge$. – Pietro Majer May 8 '14 at 18:57