Asymptotic bounds for permutations containing many different patterns

We say that a permutation σ ∈ S n contains a permutation π ∈ S k as a pattern if some subsequence of σ has the same order relations among its entries as π. We improve on results of Wilf, Coleman, and Eriksson et al. that bound the asymptotic behavior of pat ( n ) , the maximum number of distinct patterns of any length contained in a single permutation of length n. We prove that 2 n − O ( n 2 2 n − 2 n ) ⩽ pat ( n ) ⩽ 2 n − Θ ( n 2 n − 2 n ) by estimating the amount of redundancy due to patterns that are contained multiple times in a given permutation. We also consider the question of k-superpatterns, which are permutations that contain all patterns of a given length k. We give a simple construction that shows that L k , the length of the shortest k-superpattern, is at most k ( k + 1 ) 2 . This may lend evidence to a conjecture of Eriksson et al. that L k ∼ k 2 2 .