Permuton limits for some permutations avoiding a single pattern

TL;DR

The paper addresses the problem of identifying permuton limits for random permutations drawn from pattern-avoiding classes that are in bijection with avoiding an increasing pattern of length $d+1$, proving that the limit is the anti-diagonal permuton $\mu_J$ supported on the line $x+y=1$. The authors develop a BWX-based strategy to transfer structure across related pattern-avoiding classes, analyze frozen regions created by the bijections, and apply weak-convergence lemmas to establish convergence to $\mu_J$. Key contributions include a one-sided lemma controlling mass in off-diagonal regions, a detailed structure theory for permutations in these classes, and a rigorous demonstration that uniform samples converge in distribution to $\mu_J$ (with potential extensions to related classes and the anti-diagonal limit). These results offer a concrete limiting object for a broad family of pattern-avoiding classes and illustrate a framework potentially applicable to other single-pattern families and higher-length patterns.

Abstract

Permutons are probability measures on the unit square with uniform marginals that provide a natural way to describe limits of permutations. We are interested in the permuton limits for permutations sampled uniformly from certain pattern-avoiding classes that are in bijection with the class of permutations avoiding the increasing pattern of length $d+1$. In particular, we will look at a family of permutations whose permuton limit collapses to the unique permuton supported on the line $x + y = 1$ in the unit square, informally known as the anti-diagonal. We prove some general properties about permutons to aid our efforts, which may be useful for proving permuton limits that converge to the anti-diagonal for a broader range of permutation classes.

Figures (5)

• Figure 1: The traversal on the left avoids the pattern $21$ as any two points in decreasing order (like those highlighted) is not contained in a rectangle within the bounding Young diagram. The traversal on the right contains copies of the pattern $21$ (some of which are highlighted).
• Figure 2: Illustration of coloring procedure in proof of Proposition 2.1 and the Backelin-West-Xin bijection with $\sigma = I_2,$$\sigma' = J_2$ and $\tau = I_2$. The white region of the square is called the frozen region.
• Figure 3: An example of a partition of $H_i = H_i^-\bigcup H_i^\circ\bigcup H_i^+$ and $V_i = V_i^+ \bigcup V_i^\circ \bigcup V_i^-$. The blue shaded region represents $W^+_\delta$ while the pink shaded region is $W^-_\epsilon$. Note that $V_i^+ \subset W^+_\delta$, $V_i^- \subset \bigcup_{j=i+1}^{m-2} H^-_j$, and $W^-_\epsilon \subset \bigcup_{i=0}^{m-2} H_i^-$.
• Figure 4: A permutation, $\sigma \in \textbf{Av}_n(I_5)$, and its image under the Backelin-West-Xin bijection, $\rho \in \textbf{Av}_n(J_2\oplus I_3)$, with region $\text{SW}(A^2_\sigma,\sigma)$ outlined by solid blue line. The frozen region under this bijection that is not shaded can properly contain the complement of $\text{SW}(A^2_\sigma,\sigma)$ as the two lowest brown shaded points of $A^1_\sigma$ do not have a copy of $I_3$ north west of those squares.
• Figure 5: The left picture is the reverse complement of $\rho$ from Fig. \ref{['figAB']}, $\rho^{rc}\in \textbf{Av}_n(I_{3}\oplus J_{2})$. The purple curve delineates the boundary of $\text{SW}(B^{2}_{\rho^{rc}},\rho^{rc})$. The frozen region is contained in the complement of the purple shaded region and does contain some points in the complement of $\text{SW}(B^2_{\rho^{rc}},\rho^{rc})$. The right picture is the image of $\rho^{rc}$ in $\textbf{Av}_n(J_2 \oplus I_1 \oplus J_2)$ under the BWX bijection.

Theorems & Definitions (14)

• Theorem 1.1
• Proposition 2.1: BWX
• Proposition 2.2: Backelin-West-Xin Bijection
• Lemma 3.1: One-sided lemma
• proof
• Lemma 3.2
• proof
• Proposition 4.1
• proof
• Lemma 4.2