Tiling Models for Palindromic and n-Colored Compositions With 1 s and 3 s

In this work, we explore compositions of positive integers into parts consisting of 1 and 3 s, highlighting their close connection to the Narayana numbers. We also examine palindromic compositions of this type and introduce arithmetic functions that reflect their structural properties. Furthermore, the Woon and PI tree structures related to the Narayana numbers are constructed and visually illustrated. To provide an intuitive understanding, we present tiling models corresponding to these compositions and use them to derive a variety of combinatorial identities. Moreover, we extend our approach to tiling models associated with Narayana polynomials, establishing additional identities through this framework. We also introduce an n-color extension of these compositions, exploring how the additional coloring dimension affects their combinatorial properties. Finally, we consider compositions with 1 and 3 s and derive several related identities that further illustrate their relationship with the Narayana numbers.