On Trees and Noncrossing Partitions

We give a simple and natural proof of (an extension of) the identity P(k, l, n) = P2(k − 1, l − 1, n − 1). The number P(k, l, n) counts noncrossing partitions of {1, 2, . . . , l} into n parts such that no part contains two numbers x and y, 0 < y − x < k. The lower index 2 indicates partitions with no part of size three or more. We use the identity to give quick proofs of the closed formulae for P(k, l, n) when k is 1, 2, or 3. Keywords noncrossing partition, enumeration, tree, bijection.

• Type: Journal
• Year: 1998
• Where: DAM
• Authors: Martin Klazar
• Keywords: Bijection, DAM 1998, Natural Proof, Tree