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After a brief introduction to the field of pattern avoiding permutations, we describe shape results for the class of permutations avoiding the monotone and skew monotone patterns respectively. Permutations avoiding the monotone pattern [k+1,... 2,1] can be viewed as the union of k increasing sequences. We show that with high probability, each sequence must lie close to the diagonal (when viewing the permutation matrix). First, we describe an injective map from the set of permutations avoiding the given pattern to pairs of sequences, and then use probabilistic techniques to analyze the more general pairs of sequences. Our results then follow by conditioning that these sequences correspond to permutations avoiding the given pattern. We also obtain similar results for the class of permutations avoiding the skew-monotone pattern, by using a bijection of Backelin, West, and Xin. This is joint work with Erik Slivken.