Bounded affine permutations II. Avoidance of decreasing patterns

TL;DR

This paper studies bounded affine permutations avoiding decreasing patterns, providing asymptotic counts and describing their scaling limits as a union of random lines, extending combinatorial and probabilistic understanding.

Contribution

It introduces new asymptotic formulas for the enumeration of bounded affine permutations avoiding decreasing patterns and characterizes their scaling limits as a union of random lines.

Findings

Asymptotic count of permutations avoiding decreasing patterns

Explicit formulas for specific pattern avoidance cases

Description of the scaling limit as random lines

Abstract

We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size $N$ that avoid the monotone decreasing pattern of fixed size $m$. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $N\to\infty$. For instance, the number of bounded affine permutations of size $N$ that avoid $321$ is asymptotically equal to $4^N (N/4\pi)^{1/2}$. We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding $m\cdots1$ looks like $m-1$ random lines of slope $1$ whose $y$ intercepts sum to $0$.