Noncrossing linked partitions and large (3,2)(3,2)-Motzkin paths

Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and (3,2)(3,2)-Motzkin paths, where a (3,2)(3,2)-Motzkin path can be viewed as a Motzkin path for which there are three kinds of horizontal steps and two kinds of down steps. A large (3,2)(3,2)-Motzkin path is a (3,2)(3,2)-Motzkin path for which there are only two kinds of horizontal steps on the xx-axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of {1,…,n+1}{1,…,n+1} and the set of large (3,2)(3,2)-Motzkin paths of length nn, which leads to a simple explanation of the well-known relation between the large and the little Schröder numbers.