Stanley-Wilf limits are typically exponential

For a permutation π, let Sn(π) be the number of permutations on n letters avoiding π. Marcus and Tardos proved the celebrated Stanley-Wilf conjecture that L(π) = limn→∞ Sn(π) 1/n exists and is finite. Backed by numerical evidence, it has been conjectured by many researchers over the years that L(π) = Θ(k) for every permutation π on k letters. We disprove this conjecture, showing that L(π) = 2 Θ(1) for almost all permutations π on k letters.