Analysis reveals new identities for narayana numbers in tiling contexts, highlighting generalized and k-narayana variations.
We interpret the Narayana numbers combinatorially by having them count the number of tilings of an n-strip using squares and triominoes. Using this interpretation, we develop a collection of identities satisfied by the sequence of Narayana numbers. Additionally, these techniques are used to introduce the generalized Narayana numbers and the k-Narayana numbers and to prove corresponding identities.
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Plaza et al. (2025) studied this question.
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