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{\bf Maximillian M. Murphy and Vincent R. Vatter}
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{\bf Profile Classes and Partial Well-Order for Permutations}
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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 $0/\pm1$ 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 Ru\v{s}kuc. 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.
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