Noncrossing linked partitions and large -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 ) -Motzkin paths, where a ( 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 ) -Motzkin path is a ( 3 , 2 ) -Motzkin path for which there are only two kinds of horizontal steps on the x -axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of { 1 , … , n + 1 } and the set of large ( 3 , 2 ) -Motzkin paths of length n , which leads to a simple explanation of the well-known relation between the large and the little Schröder numbers.