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

Ira M. Gessel; Jang Soo Kim

doi:10.1016/j.disc.2010.07.017

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

Ira M. Gessel, Jang Soo Kim

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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.

Original language	English
Pages (from-to)	3421-3425
Number of pages	5
Journal	Discrete Mathematics
Volume	310
Issue number	23
DOIs	https://doi.org/10.1016/j.disc.2010.07.017
State	Published - 6 Dec 2010
Externally published	Yes

Keywords

Continued fraction
Dyck path
Fibonacci number
Motzkin path
Schrder path

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10.1016/j.disc.2010.07.017

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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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AU - Gessel, Ira M.

AU - Kim, Jang Soo

PY - 2010/12/6

Y1 - 2010/12/6

N2 - 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.

AB - 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.

KW - Continued fraction

KW - Dyck path

KW - Fibonacci number

KW - Motzkin path

KW - Schrder path

UR - https://www.scopus.com/pages/publications/77957282295

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DO - 10.1016/j.disc.2010.07.017

M3 - Article

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SN - 0012-365X

VL - 310

SP - 3421

EP - 3425

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 23

ER -

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

Abstract

Keywords

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