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, inequalities in the sizes of sets of pattern-avoiding permutations in this context arise from further extensions of shape-equivalence type enumerations.Definition 3.1. Let Y be a Young diagram with k rows. If A and D are disjoint subsets of [k − 1] such that if i ∈ A ∪ D, then the ith and (i + 1)st rows of Y have the same length, then we call the triple Y = (Y, A, D) an AD-Young diagram. We call Y the Young diagram of Y, A the required ascent set of Y, and D the required descent set of Y. See Figure 2.As in [1,2,14], a transversal of Young diagram Y is a set of squares T = {(i, t i )} such that every row and every column of Y contains exactly one member of T .We call Asc(T ) the ascent set of T and Des(T ) the descent set of T . If A ⊆ A ′ and D ⊆ D ′ , then we say that T a valid transversal of Y. Example 3.3. If T is a transversal of a Young diagram Y , then T is a valid transversal of the AD-Young diagram (Y, ∅, ∅).Except for a brief digression in Section 7, we restrict ourselves to the AD-Young analogues of alternating and reverse alternating permutations. Definition 3.4. Given positive integers x, y and an AD-Young diagram (Y, A, D) such that Y has k rows, we say that (Y, A, D) is x, y-alternating if A, D satisfy the property that if x − 1 ≤ i ≤ k − y, then i ∈ A if and only if i + 1 ∈ D.Example 3.5. Let Y = (4 4 ). Then, (Y, {1}, {2}) is 1-alternating, while (Y, {1, 3}, {2}) is 2-alternating but not 1-alternating. Furthermore, (Y, {2, 4}, {1, 3}) is 1-semialternating but not y-alternating for y ≤ 4.The notion of pattern avoidance is exactly as in [1,2,14]; if a transversal T = {(i, t i )} of a Young diagram Y contains a r×r permutation matrix M if there are rows a 1 < a 2 < · · · < a r and columns b 1 < b 2 < · · · < b r of Y such that (a r , b r ) ∈ Y and the restriction of T to the rows a i and the columns b i has 1's exactly where M has 1's. If T does not contain M, then we say that T avoids M. See Figure 3. Given an AD-Young diagram Y and a permutation matrix M, let S Y (M) denote the set of valid transversals of Y that avoid M. Definition 3.6. If M and N are permutation matrices such that |S Y (M)| = |S Y (N)| for all x-alternating AD-Young diagrams Y, we say that...
