2019/10/19 by Garrett, Kristina, Killpatrick, Kendra
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1910.08855
The Lucas polynomials, \n\, are polynomials in s and t given by \ n \ = s \ n-1 \ + t \ n-2 \ for n ≥ 2 with \ 0 \ = 0 and \ 1 \ = 1. The lucanomial coefficients, an analogue of the binomial coefficients, are given by \ n
k \ = \frac \n\! \k\! \n-k\!. When s = t = 1 then \ n \ = Fn and the lucanomial coefficient becomes the fibonomial coefficient \binomnkF = \fracFn!Fk! Fn-k!. The well-known Narayana numbers, Nn,k satisfy the equation Nn,k = (1)/(n) \binomnk \binomnk-1. %Cn = ∑k=1n Nn,k. % In 2018, Bennett, Carrillo, Machacek and Sagan defined the generalized Narayana numbers and conjectured that these numbers are positive integers for n ≥ 1. In this paper we define the FiboNarayana number Nn,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 ≥ 1.