Notes on Fibonacci Partitions

Let $f_1=1,f_2=2$ and $f_i=f_{i-1}+f_{i-2}$ for $i>2$ be the sequence of Fibonacci numbers. Let ${\mathrm{\Phi }}_h(n)$ be the quantity of partitions of natural number $n$ into $h$ different Fibonacci numbers. In terms of Zeckendorf partition of $n$ I deduce a formula for the function $\Phi(n;t):=\sum_{h\geq 1}\Phi_h(n)t^h$,anduseittoanalyzethefunctions$F(n):=\Phi(n;1)$and$\chi(n):=\Phi(n;-1)$\mathrm{.}Iobtaintheleastupperboundfor$F(n)$when$f_i-1<n<f_{i+1}-1$\mathrm{.}Itimpliesthat$F(n)<\sqrt{n+1}$foranynatural$n$. I prove also that $\mathrm{\mid }\chi (n)\mathrm{\mid }\text{<}1$, and $\mathop{\lim}\limits_{N\to\infty}\frac{1}{N} \left(\chi^2(1)+\chi^2(2)\dpt\chi^2(N)\right)=0$. For any $k2$, I define a special finite set $\mathbb{G}(k)$ of solutions of the equation $F(n)=k$, all solutions can be easily obtained from $\mathbb{G}(k)$. This construction uses a representation of rational numbers as certain continued fractions and provides with a canonical identification ${\coprod }_{k2}\mathbb{G}(k)={\text{G}}_{+}$, where ${\text{G}}_{+}$ is the monoid freely generated by the positive rational numbers $<1$. Let $\mathrm{\Psi }(k)$ be the cardinality of $\mathbb{G}(k)$. I prove that, for $i2k$ and $k2$, the interval $[f_i-1,f_{i+1}-1]$ contains exactly $2\mathrm{\Psi }(k)$ solutions of the equation $F(n)=k$ and offer a formula for the Dirichlet generating function of the sequence $\mathrm{\Psi }(k)$. I formulate conjectures on the set of minimal solutions of the equations $F(n)=k$ as $k$ varies and pose some questions concerning such solutions.