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A Note on Combinatorial Proofs for Extended Touchard’s and Extended Koshy’s Identities

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Abstract

Touchard’s and Koshy’s identities are beautiful identities about Catalan numbers. It is worth noting that combinatorial interpretations for extended Touchard’s identity and extended Koshy’s identity can intuitively reflect the equations. In this paper, we give a new combinatorial proof for the extended Touchard’s identity by means of Dyck Paths. The principle of inclusion–exclusion (sieve method) is employed to prove the extended Koshy’s identity. Meanwhile, as an new extension of the extended Koshy’s identity, a nice general identity is also provided.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Hebei, 066004, People’s Republic of China

    Roberta R. Zhou, Kun Yan, Yihui He & Xinxin Xu

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  1. Roberta R. Zhou
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  2. Kun Yan
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  3. Yihui He
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  4. Xinxin Xu
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Correspondence to Roberta R. Zhou.

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Hebei (A2019501024) and NSFC (11501090)

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Zhou, R.R., Yan, K., He, Y. et al. A Note on Combinatorial Proofs for Extended Touchard’s and Extended Koshy’s Identities. Graphs and Combinatorics 36, 1705–1711 (2020). https://doi.org/10.1007/s00373-020-02195-4

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  • DOI: https://doi.org/10.1007/s00373-020-02195-4

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