  We investigate pattern avoidance in alternating permutations and
generalizations thereof. First, we study pattern avoidance in an alternating
analogue of Young diagrams. In particular, we extend Babson-West's notion of
shape-Wilf equivalence to apply to alternating permutations and so generalize
results of Backelin-West-Xin and Ouchterlony to alternating permutations.
Second, we study pattern avoidance in the more general context of permutations
with restricted ascents and descents. We consider a question of Lewis regarding
permutations that are the reading words of thickened staircase Young tableaux,
that is, permutations that have (k - 1) ascents followed by a descent, followed
by (k - 1) ascents, et cetera. We determine the relative sizes of the sets of
pattern-avoiding (k - 1)-ascent permutations in terms of the forbidden pattern.
Furthermore, we give inequalities in the sizes of sets of pattern-avoiding
permutations in this context that arise from further extensions of
shape-equivalence type enumerations.
