  For a given permutation $\tau$, let $P_N^{\tau}$ be the uniform probability
distribution on the set of $N$-element permutations $\sigma$ that avoid the
pattern $\tau$. For $\tau=\mu_k:=123\cdots k$, we consider
$P_N^{\mu_k}(\sigma_I=J)$ where $I\sim \gamma N$ and $J\sim \delta N$ for
$\gamma,\delta \in (0,1)$. If $\gamma+\delta\neq 1$ then we are in the large
deviations regime with the probability decaying exponentially, and we calculate
the limiting value of $P_N^{\mu_k}(\sigma_I=J)^{1/N}$. We also observe that for
$\tau = \lambda_{k,\ell} := 12\ldots\ell k(k-1)\ldots(\ell+1)$ and
$\gamma+\delta<1$, the limit of $P_N^{\tau}(\sigma_I=J)^{1/N}$ is the same as
for $\tau=\mu_k$.
