The Lucas polynomials, $\{ n\}$, are polynomials in $s$ and $t$ given by $\{ n \} = s \{ n-1 \} + t \{ n-2 \}$ for $n \geq 2$ with $ \{ 0 \} = 0$ and $\{ 1 \} = 1$. The lucanomial coefficients, an analogue of the binomial coefficients, are given by \[ \Bigl\{ \begin{ array}{ c} n\\k \end{ array} \Bigr \} = \frac{ \{ n\}! }{ \{ k\}! \{ n-k\}!}. \] When $s = t = 1$ then $\{ n \} = F_n$ and the lucanomial coefficient becomes the fibonomial coefficient \[ \binom{ n}{ k}_F = \frac{ F_n!}{ F_k! F_{ n-k}!}. \] The well-known Narayana numbers, $N_{ n,k}$ satisfy the equation \[ N_{ n,k} = \frac{ 1}{ n} \binom{ n}{ k} \binom{ n}{ k-1}. \] \[ %C_n = \sum_{ k=1}^n N_{ n,k}. %\] In 2018, Bennett, Carrillo, Machacek and Sagan defined the generalized Narayana numbers and conjectured that these numbers are positive integers for $n \geq 1$. In this paper we define the FiboNarayana number $N_{ n,k,F}$ and give a new recurrence relation for both the FiboNarayana numbers and the generalized Narayana numbers, proving the conjecture that these are positive integers for $n \geq 1$.
Categories: math.CO