Where the monotone pattern (mostly) rules

We consider pattern containment and avoidance with a very tight definition that was used first in several papers more than 60 years ago. Using this definition, we prove that the monotone pattern is easier to avoid than almost any other pattern of the same length. We also show that with this definition, almost all patterns of length kk are avoided by the same number of permutations of length nn. The corresponding statements are not known to be true for more relaxed definitions of pattern containment. This is the first time we know of that expectations have been used to compare numbers of permutations avoiding certain patterns.