  For the elements of a Coxeter group, we present a statistic called depth,
defined in terms of factorizations of the elements into products of
reflections. Depth is bounded above by length and below by the average of
length and reflection length. In this article, we focus on the case of the
symmetric group, where we show that depth is equal to sum_i max{w(i)-i, 0}. We
characterize those permutations for which depth equals length: these are the
321-avoiding permutations (and hence are enumerated by the Catalan numbers). We
also characterize those permutations for which depth equals reflection length:
these are permutations avoiding both 321 and 3412 (also known as boolean
permutations, which we can hence also enumerate). In this case, it also happens
that length equals reflection length, leading to a new perspective on a result
of Edelman.
