Theorems and Conjectures on Some Rational Generating Functions

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 ${\sum }_{n\ge 0}{v}_2(n)x^n=(1-2x^2)/(1-2x-2x^2+2x^3)$. 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 $\mathfrak{F}$.

25 pages, two figures

Cited by in corpus (1)

• A note on Möbius functions of upho posets