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Fibonacci and Lucas analogues of binomial coefficients and what they count (Curtis Bennett, CSULB)
April 2, 2019 @ 12:15 pm - 1:10 pm
A Fibonomial is what is obtained when you replace each term of the binomial coefficients $ {n \choose k}$ by the corresponding Fibonacci number. For example, the Fibonomial
$${ 6\brace 3 } = \frac{F_6 \cdot F_5 \cdot \dots \cdot F_1}{(F_3\cdot F_2 \cdot F_1)(F_3\cdot F_2 \cdot F_1)} = \frac{8\cdot5\cdot3\cdot2\cdot1\cdot1}{(2\cdot1\cdot1)(2\cdot1\cdot1)} = 60$$
since the first six Fibonacci numbers are 1, 1, 2, 2, 5, and 8. Curiously the Fibonomials are always integers, raising the combinatorial question: what do they count? In this talk we introduce and provide a little history of the Fibonomials. We then provide a simple object the Fibonomials enumerate. We will use this new object to prove various Fibonomial analogues of standard identities on binomial coefficients and discuss further generalizations including the Lucanomials.