Abstract
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.