A Recursion for the FiboNarayana and the Generalized Narayana Numbers

TL;DR

This paper introduces FiboNarayana numbers, establishes a new recurrence relation for them and generalized Narayana numbers, and proves their positivity as integers for all n ≥ 1, extending combinatorial number theory.

Contribution

It defines FiboNarayana numbers and provides a novel recurrence relation, confirming their positivity and integrality for all relevant n, advancing understanding of these generalized combinatorial numbers.

Findings

FiboNarayana numbers are positive integers for all n ≥ 1.

A new recurrence relation for FiboNarayana and generalized Narayana numbers is established.

The conjecture of positivity and integrality is proven.

Abstract

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 $n \geq 1$.