Expected Patterns in Permutation Classes

Each length k pattern occurs equally often in the set S-n of all permutations of length n, but the same is not true in general for a proper subset of S-n. Miklos Bona recently proved that if we consider the set of n-permutations avoiding the pattern 132, all other non-monotone patterns of length 3 are equally common. In this paper we focus on the set Av(n)(123) of n-permutations avoiding 123, and give exact formulae for the occurrences of each length 3 pattern. While this set does not have the same symmetries as Av(n)(132), we find several similarities between the two and prove that the number of 231 patterns is the same in each.