Theorems and Conjectures on Some Rational Generating Functions

TL;DR

This paper studies rational generating functions derived from Fibonacci-based products, explores their properties, and introduces an associated infinite poset with combinatorial and symmetric function characteristics.

Contribution

It provides new theorems and conjectures on generating functions related to Fibonacci products and introduces an infinite poset with combinatorial and symmetric function analysis.

Findings

Explicit formula for the generating function of the sum of squares of coefficients.

Introduction of an infinite poset linked to the generating functions.

Discussion of combinatorial properties and a symmetric function encoding the poset's flag h-vector.

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 $\sum_{n\geq 0} v_2(n)x^n= (1-2x^2)/(1-2x-2x^2+2x^3)$. 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 $\mathfrak{F}$.