  We prove that the number of permutations avoiding an arbitrary consecutive
pattern of length m is asymptotically largest when the avoided pattern is
12...m, and smallest when the avoided pattern is 12...(m-2)m(m-1). This settles
a conjecture of the author and Noy from 2001, as well as another recent
conjecture of Nakamura. We also show that among non-overlapping patterns of
length m, the pattern 134...m2 is the one for which the number of permutations
avoiding it is asymptotically largest.
