# Random Periodic Pattern-Avoiding Permutations

As a fresh way to think about the probabilistic structure of pattern-avoiding permutations, we examine affine permutations with a new "boundedness" condition. We view this as an analogue of the useful concept of periodic boundary conditions in statistical physics. An *affine permutation of period* $N$ is a bijection $\omega$ of $\mathbb{Z}$ satisfying

\[\omega(i+N)=\omega(i)+N\hspace{5mm}\textrm{for every } i\in\mathbb{Z}\]

as well as the centering condition

\[\sum_{i=1}^N\omega(i)=\sum_{i=1}^N i\,,\]

and we say it is *bounded* if

\[ |\omega(i)-i| < N \hspace{5mm} \textrm{for every }i\in\mathbb{Z}. \]

Let ${\sf BA}_N$ be the set of bounded affine permutations of period $N$. Note that for any (ordinary) permutation $\sigma$ on $\{1,\ldots,N\}$, the periodic extension of $\sigma$ via $\sigma(i+kN)=\sigma(i)+kN$ ($k\in \mathbb{Z}$) is in $\textsf{BA}_N$.

For a fixed short permutation $\tau$, let ${\sf AvBA}_N(\tau)$ be the set of $\omega\in{\sf BA}_N$ that avoid the pattern $\tau$ (i.e., viewing a permutation as a sequence of integers, $\omega$ has no subsequence with the same relative order as $\tau$). We focus on the decreasing pattern $Decr_k :=\bf{k(k{-}1)\cdots 321}$ for fixed $k\geq 3$. We explain how a probabilistic viewpoint enables us to obtain the cardinality of $|{\sf AvBA}_N(Decr_k)|$ asymptotically as $N\rightarrow\infty$, and as a bonus to derive a permuton-like scaling limit for the plot of a random element of ${\sf AvBA}_N(Decr_k)$ as $N\rightarrow\infty$.

This is joint work with Justin Troyka. The research was supported in part by an NSERC Discovery Grant and by York University's Faculty of Science.