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.
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