  It is known that the set of permutations, under the pattern containment
ordering, is not a partial well-order. Characterizing the partially
well-ordered closed sets (equivalently: down sets or ideals) in this poset
remains a wide-open problem. Given a 0/+-1 matrix M, we define a closed set of
permutations called the profile class of M. These sets are generalizations of
sets considered by Atkinson, Murphy, and Ruskuc. We show that the profile class
of M is partially well-ordered if and only if a related graph is a forest.
Related to the antichains we construct to prove one of the directions of this
result, we construct exotic fundamental antichains, which lack the periodicity
exhibited by all previously known fundamental antichains of permutations.
