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In [[mathematics]], the '''Fibonomial coefficients''' or '''Fibonacci-binomial coefficients''' are defined as
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:<math>\binom{n}{k}_F = \frac{F_nF_{n-1}\cdots F_{n-k+1}}{F_kF_{k-1}\cdots F_1} = \frac{n!_F}{k!_F (n-k)!_F}</math>
 
where ''n'' and ''k'' are non-negative integers, 0&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''n'', ''F<sub>j</sub>'' is the ''j''-th [[Fibonacci number]] and ''n''!<sub>F</sub> is the ''n''th [[Fibonorial]], where 0!<sub>F</sub>, being the [[empty product]], evaluates to&nbsp;1.
 
The Fibonomial coefficients are all integers. Some special values are:
 
:<math>\binom{n}{0}_F = \binom{n}{n}_F = 1</math>
 
:<math>\binom{n}{1}_F = \binom{n}{n-1}_F = F_n</math>
 
:<math>\binom{n}{2}_F = \binom{n}{n-2}_F = \frac{F_n F_{n-1}}{F_2 F_1} = F_n F_{n-1},</math>
 
:<math>\binom{n}{3}_F = \binom{n}{n-3}_F = \frac{F_n F_{n-1} F_{n-2}}{F_3 F_2 F_1} = F_n F_{n-1} F_{n-2} /2,</math>
 
:<math>\binom{n}{k}_F = \binom{n}{n-k}_F.</math>
 
The Fibonomial coefficients {{OEIS|id=A010048}} are similar to [[binomial coefficient]]s and can be displayed in a triangle similar to [[Pascal's triangle]]. The first eight rows are shown below.
 
{| style='margin: 1em auto; text-align:center'
! width='40pt' |
! width='30pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
! width='20pt' |
|-
| <math>n=0</math> || colspan='8' | || colspan='2' | 1 || colspan ='7' |
|-
| <math>n=1</math> || colspan='7' | || colspan='2' | 1 || colspan ='2' | 1 || colspan ='6' |
|-
| <math>n=2</math> || colspan='6' | || colspan='2' | 1 || colspan ='2' | 1 || colspan ='2' | 1 || colspan ='5' |
|-
| <math>n=3</math> || colspan='5' | || colspan='2' | 1 || colspan ='2' | 2 || colspan ='2' | 2 || colspan ='2' | 1 || colspan ='4' |
|-
| <math>n=4</math> || colspan='4' | || colspan='2' | 1 || colspan ='2' | 3 || colspan ='2' | 6 || colspan ='2' | 3 || colspan ='2' | 1 || colspan ='3' |
|-
| <math>n=5</math> || colspan='3' | || colspan='2' | 1 || colspan ='2' | 5 || colspan ='2' | 15 || colspan ='2' | 15 || colspan ='2' | 5 || colspan ='2' | 1 || colspan='2' |
|-
| <math>n=6</math> || colspan='2' | || colspan='2' | 1 || colspan ='2' | 8 || colspan ='2' | 40 || colspan ='2' | 60 || colspan ='2' | 40 || colspan ='2' | 8 || colspan ='2' | 1 ||
|-
| <math>n=7</math> || colspan='1' | || colspan='2' | 1 || colspan ='2' | 13 || colspan ='2' | 104 || colspan ='2' | 260 || colspan ='2' | 260 || colspan ='2' | 104 || colspan ='2' | 13 || colspan ='2' | 1
|}
 
The recurrence relation
 
:<math>\binom{n}{k}_F = F_{n-k+1} \binom{n-1}{k-1}_F + F_{k-1} \binom{n-1}{k}_F </math>
 
implies that the Fibonomial coefficients are always integers.
 
==References==
{{reflist}}
* {{Citation | last1=Benjamin | first1=Arthur T.| last2=Plott | first2=Sean S.| authorlink2=Sean Plott |title=A combinatorial approach to Fibonomial coefficients | url=http://www.math.hmc.edu/~benjamin/papers/Fibonomial.pdf| location=Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711}}
* Ewa Krot, [http://www.springerlink.com/content/r542262451776jx6/fulltext.pdf An introduction to finite fibonomial calculus], Institute of Computer Science, Bia lystok University, Poland.
* {{MathWorld|title=Fibonomial Coefficient|urlname=FibonomialCoefficient}}
 
[[Category:Fibonacci numbers]]
[[Category:Factorial and binomial topics]]
 
 
{{Combin-stub}}

Latest revision as of 12:43, 6 November 2014

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