Noncrossing Linked Partitions and Large (3,2)-Motzkin Paths
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2010
- DOI:
- 10.48550/arXiv.1009.0176
- arXiv:
- arXiv:1009.0176
- Bibcode:
- 2010arXiv1009.0176C
- Keywords:
-
- Mathematics - Combinatorics;
- 05A15;
- 05A18
- E-Print:
- 8 pages