Title: Q-analogues of the Fibo-Stirling numbers,
Abstract: Let $F_n$ denote the $n^{th}$ Fibonacci number relative to the initial
conditions $F_0=0$ and $F_1=1$. Bach, Paudyal, and Remmel introduced Fibonacci
analogues of the Stirling numbers called Fibo-Stirling numbers of the first and
second kind. These numbers serve as the connection coefficients between the
Fibo-falling factorial basis $\{(x)_{\downarrow_{F,n}}:n \geq 0\}$ and the
Fibo-rising factorial basis $\{(x)_{\uparrow_{F,n}}:n \geq 0\}$ which are
defined by $(x)_{\downarrow_{F,0}} = (x)_{\uparrow_{F,0}} = 1$ and for $k \geq
1$, $(x)_{\downarrow_{F,k}} = x(x-F_1) \cdots (x-F_{k-1})$ and
$(x)_{\uparrow_{F,k}} = x(x+F_1) \cdots (x+F_{k-1})$. We gave a general rook
theory model which allowed us to give combinatorial interpretations of the
Fibo-Stirling numbers of the first and second kind.
There are two natural $q$-analogues of the falling and rising Fibo-factorial
basis. That is, let $[x]_q = \frac{q^x-1}{q-1}$. Then we let
$[x]_{\downarrow_{q,F,0}} = \overline{[x]}_{\downarrow_{q,F,0}} =
[x]_{\uparrow_{q,F,0}} = \overline{[x]}_{\uparrow_{q,F,0}}=1$ and, for $k > 0$,
we let $[x]_{\downarrow_{q,F,k}} = [x]_q [x-F_1]_q \cdots [x-F_{k-1}]_q$,
$\overline{[x]}_{\downarrow_{q,F,k}}= [x]_q ([x]_q-[F_1]_q) \cdots
([x]_q-[F_{k-1}]_q)$, $[x]_{\uparrow_{q,F,k}}= [x]_q [x+F_1]_q \cdots
[x+F_{k-1}]_q$, and $\overline{[x]}_{\uparrow_{q,F,k}}= [x]_q ([x]_q+[F_1]_q)
\cdots ([x]_q+[F_{k-1}]_q)$.
In this paper, we show we can modify the rook theory model of Bach, Paudyal,
and Remmel to give combinatorial interpretations for the two different types
$q$-analogues of the Fibo-Stirling numbers which arise as the connection
coefficients between the two different $q$-analogues of the Fibonacci falling
and rising factorial bases. \end{abstract}