%0 Journal Article %T A note on 2-distant noncrossing partitions and weighted Motzkin paths %A Ira M. Gessel %A Jang Soo Kim %J Mathematics %D 2010 %I arXiv %X We prove a conjecture of Drake and Kim: the number of $2$-distant noncrossing partitions of $\{1,2,...,n\}$ is equal to the sum of weights of Motzkin paths of length $n$, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial. %U http://arxiv.org/abs/1003.5301v2