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You can use the article "A bijection on Dyck paths and its consequences" by E. Deutsch. The author has several papers on enumerative problems on Dyck/Motzkin paths. (See also here)

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You can use the article "A bijection on Dyck paths and its consequences" by E. Deutsch. The author has several papers on enumerative problems on Dyck/Motzkin paths. (See also here)