A simple and direct derivation for the number of noncrossing partitions

Liaw, Sheng Chyang; Yeh, Hund‐Der; Hwang, F. K.; Chang, Gerard J.

doi:10.1090/s0002-9939-98-04546-8

Cited by 9 publications

(3 citation statements)

References 12 publications

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“…A proof of the above formula can be found in [65, e.g. corollary 9.12], see also [24,51], the pioneering article by G. Kreweras [44] and the survey of R. Simion on non-crossing partitions [81].…”

Section: Partitions and Bell Polynomialsmentioning

confidence: 99%

An Introduction to Wishart Matrix Moments

Bishop

Moral

Niclas

2018

FNT in Machine Learning

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Section: Partitions and Bell Polynomialsmentioning

confidence: 99%

An Introduction to Wishart Matrix Moments

Bishop

Moral

Niclas

2018

FNT in Machine Learning

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“…where a = n j=1 α j is the total number of parts. For a simple and direct proof of this result, see Liaw, Yeh, Hwang, and Chang [10]. In our analysis, we prove various generalizations of this result where there are further restrictions on the non-crossing partitions.…”

Section: Introductionmentioning

confidence: 70%

On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices

Cai¹,

Potechin²

2022

Preprint

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In this paper, we introduce and analyze a new operation • R which mixes two distributions Ω and Ω via a random orthogonal matrix. In particular, we take Ω • R Ω to be the limit as n → ∞ of the distribution of singular values of DRD where D and D are n × n diagonal matrices whose diagonal entries have distributions Ω and Ω respectively and R is a random n × n orthogonal matrix.We show that • R has several nice properties. We first observe that • R is commutative and associative and compute the moments of Ω • R Ω in terms of the moments of Ω and Ω . We then show that • R interacts very nicely with the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices. This allows us to answer the question posed by our previous paper of how to describe the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices when the input distribution is not {−1, 1}.In our analysis, we show that the moments of our distributions are closely connected to non-crossing partitions and prove a number of new results on non-crossing partitions which may be of independent interest.

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“…First, we begin with the number of Dyck paths of given type. Kreweras [4] shows the following theorem using recursions and Liaw et al [5] give a bijective proof.…”

Section: Dyck and Schröder Paths By Typementioning

confidence: 99%

Enumeration of Fuss-Schröder paths

Jung

Kim

2017

Electron. J. Combin.

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In this paper we enumerate the number of (k, r)-Fuss-Schröder paths of type λ. Y. Park and S. Kim studied small Schröder paths with type λ. Generalizing the results to small (k, r)-Fuss-Schröder paths with type λ, we give a combinatorial interpretation for the number of small (k, r)-Fuss-Schröder paths of type λ by using Chung-Feller style. We also give two sets of sparse noncrossing partitions of [2(k +1)n+1] and [2(k +1)n+2] which are in bijection with the set of all small and large, respectively, (k, r)-Fuss-Schröder paths of type λ.

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A simple and direct derivation for the number of noncrossing partitions

Cited by 9 publications

References 12 publications

An Introduction to Wishart Matrix Moments

An Introduction to Wishart Matrix Moments

On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices

Enumeration of Fuss-Schröder paths

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