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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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