# The Fibonacci Sequence and Schreier-Zeckendorf Sets

## TL;DR

This paper explores novel connections between Schreier sets, Zeckendorf representations, and the Fibonacci sequence, providing multiple counting methods, recurrence relations, and bijections to deepen understanding of these combinatorial structures.

## Contribution

It introduces four new ways to generate Fibonacci numbers through counting Schreier sets and establishes recurrence relations and bijections linking these sets to Fibonacci properties.

## Key findings

- C_n = F_{n+2} for weak-Schreier sets
- E_n = F_{n+2} for Zeckendorf subsets
- New recurrence relations among Schreier-Zeckendorf sets

## Abstract

A finite subset of the natural numbers is weak-Schreier if $\min S \ge |S|$, strong-Schreier if $\min S>|S|$, and maximal if $\min S = |S|$. Let $M_n$ be the number of weak-Schreier sets with $n$ being the largest element and $(F_n)_{n\geq -1}$ denote the Fibonacci sequence. A finite set is said to be Zeckendorf if it does not contain two consecutive natural numbers. Let $E_n$ be the number of Zeckendorf subsets of $\{1,2,\ldots,n\}$. It is well-known that $E_n = F_{n+2}$. In this paper, we first show four other ways to generate the Fibonacci sequence from counting Schreier sets. For example, let $C_n$ be the number of weak-Schreier subsets of $\{1,2,\ldots,n\}$. Then $C_n = F_{n+2}$. To understand why $C_n = E_n$, we provide a bijective mapping to prove the equality directly. Next, we prove linear recurrence relations among the number of Schreier-Zeckendorf sets. Lastly, we discover the Fibonacci sequence by counting the number of subsets of $\{1,2,\ldots, n\}$ such that two consecutive elements in increasing order always differ by an odd number.