Permutations that strongly avoid 132

Abstract: A permutation $π$ strongly avoids the pattern $τ$ if both $π$ and $π^2$ avoid $τ$. In this paper, we enumerate permutations of size $n$ that strongly avoid the pattern 132. This enumeration allows us to prove a conjecture that the growth rate of such permutations is 2.

• The paper introduces a novel framework that enumerates strongly 132-avoiding permutations by analyzing cycle lengths, including fixed points, transpositions, and 3-cycles.
• It leverages generating functions and Catalan numbers to derive an explicit generating function capturing all strongly 132-avoiding permutations.
• The asymptotic analysis rigorously establishes that the growth rate of these permutations is exactly 2, with subexponential corrections derived from singularity analysis.

Strong Pattern Avoidance and Enumeration: The Case of 132-Avoiding Permutations

Conceptual Framework

This paper investigates strong pattern avoidance for the permutation pattern 132. A permutation $\pi$ is said to strongly avoid the pattern $\tau$ if both $\pi$ and its square $\pi^2$ (in one-line notation) avoid $\tau$. The motivation for this study arises from questions posed by Bóna and Smith about the enumeration and growth rate of strongly 132-avoiding permutations, which were previously unresolved. The analysis of strong 132-avoidance is closely tied to the cycle structure of permutations, particularly those restricted by pattern avoidance.

Strong pattern avoidance generalizes classical permutation pattern avoidance and integrates permutation powers. Prior literature established exact results for some patterns (e.g., monotone increasing and 312), but left 132 (and equivalently 213 and 321) unresolved, with only bounds and conjectures available [BS2019, BD2020]. The relationship between cycle structure and pattern avoidance provides the analytic foundation for tackling the enumeration problem.

Enumeration Strategy and Structural Analysis

The authors develop a comprehensive enumeration framework by analyzing strongly 132-avoiding permutations according to the cycle length containing the largest element $n$. Permutations are partitioned as follows:

• $k=1$ (fixed points): Simple recursive relationship with smaller strongly 132-avoiding permutations.
• $k=2$ (transpositions): Leveraging Simion and Schmidt's involution counts, with the additional restriction of 132-avoidance [SS1985].
• $k=3$ (3-cycles): Employing results from [AG2022], exploiting the fact that 132-avoiding permutations composed solely of 3-cycles automatically strongly avoid 132, since for such permutations $\pi^2 = \pi^{-1}$.
• $\tau$0: Deep structural analysis reveals these occur only when $\tau$1 divides $\tau$2, and explicit formulas are obtained by encoding the permutation as a composition of smaller 132-avoiding permutations and precise shifts dictated by the cycle.

For each case, the enumeration reduces to sophisticated generating function manipulations. In particular, the generating function associated with Catalan numbers ($\tau$3) appears repeatedly due to the classical connection between 132-avoidance and Catalan sequences.

A central result is the main generating function: $\tau$4 This generating function captures all strongly 132-avoiding permutations of size $\tau$5 and is derived via explicit enumeration in each case.

Asymptotic Analysis and Growth Rate

Through a careful singularity analysis of the main generating function, the authors rigorously establish the asymptotic behavior of the coefficients $\tau$6 (the number of strongly 132-avoiding permutations of size $\tau$7). By expanding around the dominant singularity at $\tau$8, it is shown that

$\tau$9

with an explicit constant $\pi$0. This confirms that the growth rate is exactly 2, providing a definitive answer to Bóna and Smith's conjecture regarding the exponential growth rate. Furthermore, it is established that $\pi$1 for sufficiently large $\pi$2.

Numerical and Structural Results

The enumeration captures fine structural details:

• For $\pi$3, permutations are only possible when $\pi$4 divides $\pi$5, and the cycle containing $\pi$6 has length $\pi$7 for an appropriate $\pi$8.
• Strong claims are made regarding the cycle structure—specifically, full characterization of permutations where $\pi$9 is in cycles of length $\pi^2$0.
• The explicit enumeration via Catalan numbers for each cycle length yields exact counts, with strong bounds on the relevant cases.

Notably, the generating function reveals analytic properties such as singularities and regularity, confirming that the sequence is well-behaved and substantiating the asymptotic results.

Theoretical Implications

The results provide a unifying enumeration for a previously elusive case of strong pattern avoidance. The tight link between cycle structure and pattern avoidance is crystallized, allowing extension to broader classes of patterns and powers. The analytic techniques can be generalized to stronger notions, such as powerful avoidance (where $\pi^2$1 avoids $\pi^2$2 for all $\pi^2$3), and chain avoidance (with distinct patterns for $\pi^2$4 and $\pi^2$5).

Furthermore, the precise enumeration and asymptotics bridge cycle-type enumeration, pattern-avoiding permutation theory, and generating function analysis.

Future Directions

Potential developments include:

• Generalization to powerful and chain avoidance for longer and more complex patterns.
• Extension of analytic enumeration to other patterns and their interaction with permutation power structure.
• Algorithmic applications for sampling and counting strongly pattern-avoiding permutations.
• Connections to algebraic combinatorics, symmetric group powers, and algebraic generating function theory.

Conclusion

This paper resolves a long-standing conjecture in strong pattern avoidance, providing exact enumeration and analytic properties for permutations that strongly avoid 132. The growth rate is rigorously established as 2, with subexponential correction, and structural insights into the cycle composition are furnished. The techniques and results lay a foundation for further explorations in strong and powerful pattern avoidance, marking a substantial advancement in the combinatorial analysis of permutations (2604.25912).