A note on 2-distant noncrossing partitions and weighted Motzkin paths

We prove a conjecture of Drake and Kim: the number of 22-distant noncrossing partitions of {1,2,…,n}{1,2,…,n} is equal to the sum of weights of Motzkin paths of length nn, 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.