On the number of fixed points of the map $\gamma$

TL;DR

This paper studies a recursively defined sequence related to fixed points of a specific map, revealing its binary nature and number-theoretic properties, thereby addressing existing conjectures in the field.

Contribution

It introduces a recursive sequence linked to fixed points of a map and explores its properties, partially resolving prior conjectures.

Findings

Sequence contains only 0s and 1s

Number-theoretic properties of the sequence analyzed

Partial resolution of conjectures by Cori et al.

Abstract

We recursively define a sequence $\{F_{n,k}\}_{n,k\in\mathbb N }$ and we prove that such sequence contains only the symbols $\{0,1\}.$ We investigate some number-theoretic properties of such sequence and of the way it can be generated. The number $F_n$ can be interpreted as the number of fixed points of semilength $n$ of the map $\gamma$ introduced by Barnabei et al. Our results partially answer conjectures posed by Cori et al.