  In this paper, we show that both 12312-avoiding partitions and 12321-avoiding
partitions of the set $[n+1]$ are in one-to-one correspondence with Schr\"oder
paths of semilength $n$ without peaks at even level. As a consequence, the
refined enumeration of 12312-avoiding (resp. 12321-avoiding) partitions
according to the number of blocks can be reduced to the enumeration of certain
Schr\"oder paths according to the number of peaks. Furthermore, we get the
enumeration of irreducible 12312-avoiding (resp. 12321-avoiding) partitions,
which are closely related to skew Dyck paths.
