Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns

For $\eta \in S_3$, let $S_n^{\text{av}\left(\eta \right)}$ denote the set of permutations in $S_n$ that avoid the pattern η, and let $E_n^{\text{av}\left(\eta \right)}$ denote the expectation with respect to the uniform probability measure on $S_n^{\text{av}\left(\eta \right)}$. For $n\ge k\ge 2$ and $\tau \in S_k^{\text{av}\left(\eta \right)}$, let $N_n^{\left(k\right)}\left(\sigma \right)$ denote the number of occurrences of k consecutive numbers appearing in k consecutive positions in $\sigma \in S_n^{\text{av}\left(\eta \right)}$, and let $N_n^{(k;\tau )}\left(\sigma \right)$ denote the number of such occurrences for which the order of the appearance of the k numbers is the pattern τ. We obtain explicit formulas for $E_n^{\text{av}\left(\eta \right)}{N}_n^{(k;\tau )}$ and $E_n^{\text{av}\left(\eta \right)}{N}_n^{\left(k\right)}$, for all $2\le k\le n$, all $\eta \in S_3$ and all $\tau \in S_k^{\text{av}\left(\eta \right)}$. These exact formulas then yield asymptotic formulas as $n\to \infty$ with k fixed, and as $n\to \infty$ with $k=k_n\to \infty$. We also obtain analogous results for $S_n^{\text{av}({\eta }_1,\cdots ,{\eta }_r)}$, the subset of $S_n$ consisting of permutations avoiding the patterns ${\left\{{\eta }_i\right\}}_{i=1}^r$, where ${\eta }_i\in S_{m_i}$, in the case that ${\left\{{\eta }_i\right\}}_{i=1}^n$ are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to $r=2$, ${\eta }_1=2413,{\eta }_2=3142$.