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Theorems and Conjectures on Some Rational Generating Functions (2101.02131v3)
Published 6 Jan 2021 in math.CO
Abstract: Let $I_n(x)=\prod_{i=1}n \left( 1+x{F_{i+1}}\right)$, where $F_{i+1}$ denotes a Fibonacci number. Let $v_r(n)$ denote the sum of the $r$th powers of the coefficients of $I_n(x)$. Our prototypical result is that $\sum_{n\geq 0} v_2(n)xn= (1-2x2)/(1-2x-2x2+2x3)$. We give many related results and conjectures. A certain infinite poset $\mathfrak{F}$ is naturally associated with $I_n(x)$. We discuss some combinatorial properties of $\mathfrak{F}$ and a natural generalization, including a symmetric function that encodes the flag $h$-vector of $\mathfrak{F}$.