UCLA

Following the techniques initiated in \cite{MP}, we continue to study the limit

shapes of random permutations avoiding a specific subset of patterns.

We consider patterns in $S_3$ extensively, and also prove some results

regarding pairs of permutations with one in $S_3$ and another in $S_4$.

We analyze the limiting distribution of a pattern-avoiding permutation,

and calculate the asymptotic behavior of distributions of positions of numbers

in the permutations. The distributions vary significantly depending on

which patterns are avoided. We also apply our results to obtain results on various

permutation statistics.