Zigzag Partitions And Fibonacci Sequence Of Odd Indexes

In this paper, we introduce zigzag partitions, which is closely related to m-regular partitions and RNA secondary structures. By applying Chen-Deng-Du's reduction algorithm, we present the generating function of zigzag partitions with variables marking the number of vertices, arcs and loops. As a consequence, the zigzag partitions with a fixed number of vertices, arcs and loops are enumerated by the binomial coefficients. Furthermore, the zigzag partitions with a fixed number of vertices and arcs are counted by the Fibonacci sequence of odd indexes. Finally, a bijection between zigzag partitions and nondecreasing Dyck paths is established.