**Title**: Consecutive patterns in permutations

**Abstract: **An occurrence of a consecutive pattern sigma in a permutation pi

is a subsequence of adjacent entries of pi in the same relative

order as the entries of sigma. For example, occurrences of the

consecutive pattern 21 are descents, and alternating permutations

are those that avoid the consecutive patterns 123 and 321. The

systematic study of consecutive patterns in permutations started over

a decade ago. More recently, consecutive patterns have become

relevant in the study of one-dimensional dynamical systems, and they

are useful in creating tests to distinguish random from deterministic

time series.

We show that the number of permutations avoiding the consecutive

pattern 12…m —that is, containing no m adjacent entries in

increasing order— is asymptotically larger than the number of

permutations avoiding any other consecutive pattern of length m. At

the other end of the spectrum, the number of permutations avoiding

12… (m-2)m(m-1) is asymptotically smaller than for any other

consecutive pattern.