  We study properties of the forgotten monoid which appeared in work of Lascoux
and Schutzenberger and recently resurfaced in the construction of dual
equivalence graphs by Assaf. In particular, we provide an explicit
characterization of the forgotten classes in terms of inversion numbers and
show that there are n^2-3n+4 forgotten classes in the symmetric group S_n. Each
forgotten class contains a canonical element that can be characterized by
pattern avoidance. We also show that the sum of Gessel's quasi-symmetric
functions over a forgotten class is a 0-1 sum of ribbon-Schur functions.
