Combinatorial Proofs of Some Formulas for Triangular Tilings

Bodeen et al. recently considered a new combinatorial tiling problem wherein a “strip ” is tiled using triangles of four types and derived various identities for the re-sulting numbers. Some of the identities were proven combinatorially and others only algebraically, and the question of finding combinatorial interpretations of all of their results was posed. In this note, we provide the requested bijective proofs. To do so, we rephrase the question in an equivalent form in terms of tiling a strip with squares, trominos, and three types of dominos and form bijections or near bijections where the cardinality of various size families of sets gives the correct result.