# Direct bijections for $s,t$-Fibonomial identities

Sagan and Savage gave a combinatorial interpretation of a polynomial generalization of Fibonomial coefficients. Their proof uses the recurrence relation for the Lucas polynomials that generalize the Fibonacci numbers, namely $\{0\}=0$, $\{1\}=1$, $\{n\}=s\{n-1\}+t\{n-2\}$ for $n\ge 2$. Define $s$ as a weight of a monomino, $t$ is a weight of a domino, and extend the weight function multiplicatively as usual. Then the $s,t$-Fibonacci number $\{n+1\}$ is the sum of all possible weights of an $n\times 1$ strip of cells. Subsequently, $\{n\}!=\{n\}\{n-1\}\dots\{1\}$, $\{0\}!=1$, and the $s,t$-Fibonomial coefficient $${m+n \brace m}=\frac{\{m+n\}!}{\{m\}!\{n\}!}$$ is the sum of weights of fillings of an $m\times n$ rectangle by pairs of complementary partitions $(\lambda,\lambda^*)$ so that rows of $\lambda$ and columns of $\lambda^*$ are filled by monominos and dominos, and every column of $\lambda^*$ starts with a domino (see Figure 2 in the paper).

The proof that ${m+n \brace n}$ is a combinatorial interpretation of the $s,t$-Fibonomial coefficients is recursive, using the identity $${m+n \brace m}=\{n+1\}{m+n-1 \brace m-1} + t\{m-1\}{m+n-1 \brace m}.$$

However, I am wondering if there have since been any direct bijections, using this combinatorial interpretation of $s,t$-Fibonomials, corresponding to the following identities: $$\begin{split} {m+n \brace m}&={m+n \brace n}\\ {m+n+1 \brace m+1}\{m+1\}&={m+n+1 \brace m}\{n+1\}={m+n \brace m}\{m+n+1\}\\ {m+n \brace m}\{m\}!&=\{m+n\}\{m+n-1\}\dots\{m+1\} \end{split}$$

• Thanks, Bruce, and welcome to MO! I actually started by thinking about FiboCatalan polynomials and then realized that to prove $\frac{1}{\{n+1\}}{2n \brace n}={2n-1 \brace n-1}+t{2n-1 \brace n-2}$ combinatorially from your recurrence, I would need to show $\{n-1\}{2n-1 \brace n}=\{n+1\}{2n-1 \brace n+1}$. This implied looking at the identities I listed. – Alexander Burstein Jan 11 '18 at 21:12