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

TL;DR

This paper proves a conjecture linking 2-distant noncrossing partitions to weighted Motzkin paths, establishing a new combinatorial connection involving Fibonacci numbers through two different proof methods.

Contribution

It confirms a conjecture by Drake and Kim, providing both a continued fraction and a combinatorial proof of the relationship.

Findings

Number of 2-distant noncrossing partitions equals sum of weighted Motzkin paths

Weights involve products of fractions with Fibonacci numbers

Two distinct proofs validate the conjecture

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.