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

I. A. Pushkarev

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part I, Tome 223 (1995), pp. 280-312

Résumé

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.