  Permutations that avoid given patterns are among the most classical objects
in combinatorics and have strong connections to many fields of mathematics,
computer science and biology. In this paper we study the scaling limits of a
random permutation avoiding a pattern of length 3 and their relations to
Brownian excursion. Exploring this connection to Brownian excursion allows us
to strengthen the recent results of Madras and Pehlivan, and Miner and Pak as
well as to understand many of the interesting phenomena that had previously
gone unexplained.
