123-Avoiding permutations with an adjacency constraint

We study permutations $\pi \in S_n$ that avoid $123$ while satisfying the adjacency constraint $\mid {\pi }_{i+1}-{\pi }_i\mid \le m$ 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 restriction imposed by the interaction between pattern avoidance and bounded adjacency. For every fixed $m$, we then construct an exact finite-state description of the family, with precisely $2^{m+1}-m-1$ realizable states. This yields an explicit transfer-matrix enumeration and, in particular, a rational generating function for every fixed $m$; consequently, the enumeration sequence satisfies an eventual constant-coefficient linear recurrence. We further identify the exponential growth rate with the spectral radius ${\rho }_m$ of the corresponding transfer matrix and show that ${\rho }_m<4$, is nondecreasing in $m$, and converges to the Catalan growth constant $4$ as $m\to \mathrm{\infty }$. Finally, we show that the bounded-adjacency constraint breaks the ordinary length-three Wilf equivalence, and we develop algorithmic and graph-theoretic interpretations of the family, including a correspondence with constrained Hamiltonian paths in powers of the path graph.