  Consider the Fibonacci numbers defined by setting $F_1=1=F_2$ and $F_n
=F_{n-1}+F_{n-2}$ for $n \geq 3$. We let $n_F! = F_1 \cdots F_n$ and
$\binom{n}{k}_F = \frac{n_F!}{k_F!(n-k)_F!}$. Let $(x)_{\downarrow_0} =
(x)_{\uparrow_0} = 1$ and for $k \geq 1$, $(x)_{\downarrow_k} = x(x-1) \cdots
(x-k+1)$ and $(x)_{\uparrow_k} = x(x+1) \cdots (x+k-1)$. Then the Stirling
numbers of the first and second kind are the connections coefficients between
the usual power basis $\{x^n:n \geq 0\}$ and the falling factorial basis
$\{(x)_{\downarrow_n}:n \geq 0\}$ in the polynomial ring $\mathbb{Q}[x]$ and
the Lah numbers are the connections coefficients between the rising factorial
basis $\{(x)_{\uparrow_n}:n \geq 0\}$ and the falling factorial basis
$\{(x)_{\downarrow_n}:n \geq 0\}$ in the polynomial ring $\mathbb{Q}[x]$.
  The goal of this paper is to find Fibonacci analogues for the Stirling
numbers of the first and second kind and the Lah numbers. Our idea is to
replace the falling factorial basis and the rising factorial basis by the
Fibo-falling factorial basis $\{(x)_{\downarrow_{F,n}}:n \geq 0\}$ and the
Fibo-rising factorial basis $\{(x)_{\uparrow_{F,n}}:n \geq 0\}$ where
$(x)_{\downarrow_{F,0}} = (x)_{\uparrow_{F,0}} = 1$ and for $k \geq 1$,
$(x)_{\downarrow_{F,k}} = x(x-F_1) \cdots (x-F_{k-1})$ and
$(x)_{\uparrow_{F,k}} = x(x+F_1) \cdots (x+F_{k-1})$. Then we study the
combinatorics of the connection coefficients betweenthe usual power basis, the
Fibo-falling factorial basis, and the Fibo-rising factorial basis. In each
case, we can give a rook theory model for the connections coefficients and show
how this rook theory model can give combinatorial explanations for many of the
properties of these coefficients.
