123-Avoiding permutations with an adjacency constraint

We study permutations $π\in S_n$ that avoid $123$ while satisfying the adjacency constraint $\lvert π_{i+1}-π_i\rvert \le m$ for some $m\in\mathbb{Z}^{+}$. We first show that every admissible permutation is localized within vertical distance $m$ of the reverse diagonal, revealing a strong global res