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 $\{(x{)}_{{\downarrow }_{F,n}}:n\ge 0\}$ and the Fibo-rising factorial basis $\{(x{)}_{{\uparrow }_{F,n}}:n\ge 0\}$ which are defined by $(x{)}_{{\downarrow }_{F,0}}=(x{)}_{{\uparrow }_{F,0}}=1$ and for $k\ge 1$, $(x{)}_{{\downarrow }_{F,k}}=x(x-F_1)\cdots (x-F_{k-1})$ 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}

arXiv admin note: substantial text overlap with arXiv:1510.04310