Q-analogues of the Fibo-Stirling numbers (1701.07515v1)

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}