Binary operations on pattern-avoiding cycles

TL;DR

This work introduces partial binary operations on pattern-avoiding cycles to build larger cyclic permutations from smaller ones, enabling recursive lower bounds for the counts $c_n(σ)$ of cyclic $n$-permutations avoiding patterns $σ$ of length three. By analyzing four symmetry-reduced cases—$123$, $132$, $312$, and $321$—the authors derive explicit construction rules and disjoint counting families that yield substantial growth-rate lower bounds: at least $3$ for $σ∈\{231,312,321\}$ and at least $2.6$ for $σ∈\{123,132,213\}$, with several sections improving known bounds such as proving $c_n(σ)\ge 2c_{n-1}(σ)$ for many $σ$ and $n$. The approach blends combinatorial decompositions, operator-based assembly, and Dyck-path techniques to analyze the structure of pattern-avoiding cyclic permutations, offering a roadmap toward sharper enumeration and highlighting open questions in the cyclic-avoidance regime. The results provide both concrete recursive lower bounds and insights into the intricate interplay between pattern-avoidance and cyclic structure in permutations.

Abstract

Suppose $c_n(σ)$ denotes the number of cyclic permutations in $\mathcal{S}_n$ that avoid a pattern $σ$. In this paper, we define partial groupoid structures on cyclic pattern-avoiding permutations that allow us to build larger cyclic pattern-avoiding permutations from smaller ones. We use this structure to find recursive lower bounds on $c_n(σ)$. These bounds imply that $c_n(σ)$ has a growth rate of at least 3 for $σ\in\{231,312,321\}$ and a growth rate of at least 2.6 for $σ\in\{123,132,213\}$. In the process, we prove (and sometimes improve) a conjecture of Bóna and Cory that $c_n(σ)\geq 2 c_{n-1}(σ)$ for all $σ\in\mathcal{S}_3\setminus\{123\}$ and $n\geq 2.$

Figures (6)

• Figure 1: This example demonstrates the operation defined in Definition \ref{['def:312-star']}. Here, $\alpha$ ends in 21. Take all the elements before that 21 and append them to the front of $\beta$ to obtain $\alpha*\beta.$ The result is still cyclic and 312-avoiding.
• Figure 2: This example demonstrates the operation defined in Definition \ref{['defn:321-combine']}. Here $\alpha\in\mathcal{C}_7(321)$ and $\beta\in\mathcal{C}_8(321)$. In this case, by Lemma \ref{['lem:321-all']}, $\alpha\odot\beta$ is still cyclic and 321-avoiding since $\alpha_7\neq 6$ and $\beta_2\neq 1$.
• Figure 3: This example demonstrates the operation defined in Definition \ref{['defn:123']}. Here, $\beta\in\mathcal{C}_6(123)$ and $\alpha\in\mathcal{C}_8(123)$ with the first four elements of $\alpha$ all greater than the second four elements of $\alpha$. Thus we can compute $\alpha\star\beta$, which places the large elements of $\alpha$ in front and the small elements of $\alpha$ at the end, with $\beta$ in the middle (with two of the elements from both $\alpha$ and $\beta$ "merged" to keep it a cycle). Thus $\alpha$ is "wrapped around" $\beta$ and the result is a 123-avoiding cyclic permutation of length 12.
• Figure 4: This example demonstrates the operation defined in Definition \ref{['defn:132']}. Here $\beta\in\mathcal{C}_8(132)$ and $\alpha\in\mathcal{C}_6(132)$ with the first three elements of $\alpha$ all greater than the second three elements of $\alpha$. Thus we can compute $\alpha\,$$\,\beta$, which places the large elements of $\alpha$ in front and the small elements of $\alpha$ at the end, with $\beta$ in the middle (with two of the elements from both $\alpha$ and $\beta$ "merged" to keep it a cycle). Thus $\alpha$ is "wrapped around" $\beta$ and the result is a 132-avoiding cyclic permutation of length 12.
• Figure 5: On the left, an example of a 132-avoiding permutation, $\pi = 76821345$, together with Dyck path $D(\pi)$. On the right, the permutation $p(D_L(\pi)) = 8 7 9 5 2 1 3 4 5$ associated to the Dyck path obtained by adding $ud$ in the correct place on the original Dyck path for $\pi$. The resulting permutation is still cyclic and still avoids the pattern 132.

Theorems & Definitions (39)

• Lemma 3.2
• proof
• Example 3.3
• Definition 3.4
• Example 3.5
• Lemma 3.6
• proof
• Lemma 3.7
• proof
• Theorem 3.8