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arXiv · 1009.0176

Noncrossing Linked Partitions and Large (3,2)-Motzkin Paths

William Y. C. Chen·Carol J. Wang
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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.

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William Y. C. Chen, Carol J. Wang. 2010-09-01. Noncrossing Linked Partitions and Large (3,2)-Motzkin Paths. https://arxiv.org/abs/1009.0176

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