Fibonomial coefficients

Posted on by John

Binomial coefficients can be defined by

{n \choose k} = \frac{n(n-1)(n-2)\cdots(n-k+1)}{k(k-1)(k-2)\cdots 1}

Edouard Lucas defined the Fibonomial coefficients analogously by

{n \choose k}_{\cal F} = \frac{F_nF_{n-1}F_{n-2}\cdots F_{n-k+1}}{F_kF_{k-1}F_{k-2}\cdots F_1}

where Fj is the jth Fibonacci number.

Binomial coefficients satisfy

{n \choose k} = {n-1 \choose k} + {n-1 \choose k-1}

and Fibonomial coefficients satisfy an analogous identity

{n \choose k}_{\cal F} = F_{k-1}{n-1 \choose k}_{\cal F} + F_{n-k+1}{n-1 \choose k-1}_{\cal F}

Incidentally, this identity can be used to show that Fibonomial coefficients are integers, which isn’t obvious from the definition.

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4 thoughts on “Fibonomial coefficients”

  2. John

    Not that I know of, but I just ran across it tonight, so I’m no authority.

    I imagine it has applications in combinatorics, but I’m guessing. Several people have written papers on it.

  3. David Harris

    Um… to prove that Fibonacci are integers, isn’t it easier to use the fact that they satisfy the recursion
    F_k = F_{k-1} + F_{k-2}

  4. John

    My comment wasn’t about the Fibonacci numbers but the Fibonomial coefficients. It’s obvious that Fibonacci numbers are integers, but it’s not obvious that the ratio of Fibonacci products defining the Fibonomial coefficients is always an integer.

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