The lattices of ideals of multizigzags and the enumeration of Fibonacci partitions

Let $u_1=1$, $u_2=2$, $u_3,\dots$ be the sequence of Fibonacci numbers. A Fibonacci partition of a natural number $n$ is a partition of $n$ into different Fibonacci numbers. In this paper it is proved that the set of Fibonacci partitions of a natural number, partially ordered with respect to refinement is the lattice of ideals of a multizigzag. On the basis of this theorem we obtain some results concerning the coefficients of the Taylor series of infinite products
$$ \prod_{i=1}^{+\infty}(1+zq^{u_i})=1+\sum_{k=1}^{+\infty}a_k(z)q^k, $$
where $z=\pm1$, $-\frac12\pm i\frac{\sqrt3}2$, $\pm i$. Bibliography: 6 titles.