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

Abstract

Let u₁=1, u₂=2, u₃,... 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

$$\prod\limits_{i = 1}^{ + \infty } {\left( {1 + zq^{ui} } \right) = 1 + \sum\limits_{k = 1}^{ + \infty } {a_k \left( z \right)q^k } } $$

where$z = \pm 1, - \tfrac{1}{2} \pm i\tfrac{{\sqrt 3 }}{2}$, ±i. Bibliography: 6 titles.