An Exploration of Combinatorial Interpretations for Fibonomial Coefficients

We can define Fibonomial coefficients as an analogue to binomial coefficients as F(n,k) = FnFn-1 … F­n-k+1 / F­kFk-­1…F1, where Fn represents the nth Fibonacci number. Like binomial coefficients, there are many identities for Fibonomial coefficients that have been proven algebraically. However, most of these identities have eluded combinatorial proofs. Sagan and Savage (2010) first presented a combinatorial interpretation for these Fibonomial coefficients. More recently, Bennett et al. (2018) provided yet another interpretation, that is perhaps more tractable. However, there still has been little progress towards using these interpretations of the Fibonomial coefficient to prove any of the identities. Within this thesis, I seek to explore both proofs for Fibonomial identities that have yet to be explained combinatorially, as well as potential alternatives to the thus far proposed combinatorial interpretations of the coefficients themselves