Bounded affine permutations II. Avoidance of decreasing patterns

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 $m\cdots 1$ looks like $m-1$ random lines of slope $1$ whose $y$ intercepts sum to $0$.

References in corpus (4)

• Universal limits of substitution-closed permutation classes
• A decorated tree approach to random permutations in substitution-closed classes
• Square permutations are typically rectangular
• Bounded affine permutations I. Pattern avoidance and enumeration