Finite-state enumeration of adjacency-constrained 132-avoiding permutations
For a fixed integer $m\ge 1$, let $\mathcal{A}_n^{(m)}$ be the set of permutations $\pi\in S_n$ that avoid the pattern $132$ and satisfy the adjacency bound $|\pi_{i+1}-\pi_i|\le m$ for all $i$. Here, a pattern $132$ means three indices $i<j<k$ such that $\pi_i<\pi_k<\pi_j$. A recent study initiated the enumeration of these constrained 132-avoiding permutations, treating the case $m=2$ by deriving a rational ordinary generating function and asking for finite-state decompositions, rational generating functions, and explicit rational formulas for larger fixed $m$. We introduce a two-sided endpoint-state decomposition that works uniformly for every fixed $m$. The state variables impose threshold bounds on the endpoint deficiencies $n-\pi_1$ and $n-\pi_n$, with thresholds in $\{0,1,\ldots,m-1,\infty\}$. This gives at most $(m+1)^2$ states and proves that, for every fixed $m$, the ordinary generating function $A^{(m)}(x)$ is rational and can be computed effectively by exact linear algebra. We also identify cyclic strongly connected components of the dependency graph in the finite-state system to give an explicit upper bound for the order of an eventual constant-coefficient recurrence satisfied by the sequence $a_n^{(m)}=|\mathcal{A}_n^{(m)}|$. We then recover the known case $m=2$ from this state system and work out the case $m=3$ explicitly. On the asymptotic side, we prove that the exponential growth constant exists for every $m$; for $m\ge2$ it is obtained from the spectral radii of the two cyclic components with more than one vertex in the state system. We determine the simple-pole asymptotics for $m=2$ and $m=3$, and we prove that the growth constants are nondecreasing in $m$, strictly smaller than the Catalan growth constant $4$ for every finite $m$, and converge to $4$ as $m\to\infty$.