Odd-indexed Fibonacci numbers via pattern-avoiding permutations

TL;DR

This paper explores combinatorial structures related to odd-indexed Fibonacci numbers, establishing bijections with pattern-avoiding permutations and deriving generating functions for positional statistics.

Contribution

It introduces new bijections between Fibonacci-related combinatorial objects and pattern-avoiding permutations, along with generating functions for specific statistics.

Findings

Enumerates structures linked to odd Fibonacci numbers

Establishes bijections with pattern-avoiding permutations

Derives generating functions for positional statistics

Abstract

In this paper, we consider several combinatorial problems whose enumeration leads to the odd-indexed Fibonacci numbers, including certain types of Dyck paths, block fountains, directed column-convex polyominoes, and set partitions with no crossings and no nestings. Our goal is to provide bijective maps to pattern-avoiding permutations and derive generating functions that track certain positional statistics at the permutation level.