On 132-Avoiding Permutations with an Adjacency Constraint

TL;DR

This paper investigates permutations avoiding the pattern 132 with an adjacency constraint, providing explicit solutions for the case m=2 and conjecturing a general finite-state structure for fixed m.

Contribution

It offers a complete solution for the case m=2, including recurrences and generating functions, and proposes a conjecture for the general case with fixed m.

Findings

Explicit recurrences and rational generating functions for m=2

Asymptotic growth rate approximately 1.4656 for m=2

Conjecture of finite-state decomposition for fixed m leading to growth constants approaching 4

Abstract

We study permutations in $S_n$ that simultaneously avoid the pattern $132$ and satisfy the adjacency bound $|\pi_{i+1} - \pi_i| \leq m$ for all $i$, denoting their number by $A_n^{(m)}$. This combination of a global pattern restriction and a local bounded-difference condition produces a strong structural collapse: whereas unrestricted $132$-avoiding permutations are counted by the Catalan numbers with exponential growth rate $4$, the adjacency constraint forces the maximum element $n$ to occupy only positions in $\{1, 2, \ldots, m\} \cup \{n\}$. We give a complete solution for $m = 2$ by partitioning the class according to the position of the maximum element. This yields explicit recurrences and a rational generating function, from which we derive asymptotic growth of the form $A_n^{(2)} \sim C \alpha^n$ with $\alpha \approx 1.4656$. We conjecture that for each fixed $m$, the class admits a finite-state structural decomposition leading to linear recurrences with constant coefficients and rational generating functions, with growth constants increasing to $4$.