Noncrossing Linked Partitions and Large (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 colored Motzkin paths. A (3,2)-Motzkin path can be viewed as a colored Motzkin path in the sense that there are three types of level steps and two types of down steps. A large (3,2)-Motzkin path is defined to be a (3,2)-Motzkin path for which there are only two types of level steps on the x-axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of [n+1] and the set of large (3,2)-Motzkin paths of length n. In this setting, we get a simple explanation of the well-known relation between the large and the little Schroder numbers.