Permutation Patterns of the Iterated Syracuse Function

TL;DR

This paper investigates the permutation patterns of triples generated by the Syracuse function, showing certain patterns occur with positive density while others do not, revealing structural properties of the iterated function.

Contribution

It proves the existence of positive density for all permutations of (1,2,3) and demonstrates the nonexistence of some patterns for permutations of length 4 or more.

Findings

Permutation patterns of triples have positive density.

Some permutation patterns of quadruples do not occur.

Certain permutation patterns are impossible for all n ≥ 4.

Abstract

Let $\Omega$ be the set of odd positive integers and let $S:\Omega \rightarrow \Omega$ be the Syracuse function. It is proved that, for every permutation $\sigma$ of $(1,2,3)$, the set of triples of the form $(m,S(m),S^2(m))$ with permutation pattern $\sigma$ has positive density, and these densities are computed. However, there exist permutations $\tau$ of $(1,2,3,4)$ such that no quadruple $(m,S(m), S^2(m), S^3(m))$ has permutation pattern $\tau$. This implies the nonexistence of certain permutation patterns of $n$-tuples $(m,S(m),\ldots, S^{n-1}(m))$ for all $n \geq 4$.