Permutation Patterns of the Iterated Syracuse Function

Let $\mathrm{\Omega }$ be the set of odd positive integers and let $S:\mathrm{\Omega }\to \mathrm{\Omega }$ 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 $(m,S(m),\dots ,S^{n-1}(m))$ for all $n\ge 4$.