Beyond Alternating Permutations: Pattern Avoidance in Young Diagrams and Tableaux

Abstract

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...

Ascent-Descent Young Diagrams and Pattern Avoidance in Alternating Permutations

“…This paper proves a special case of a conjecture of Gowravaram and the author [5,6]. We now review the definitions of AD-Young diagrams and the alternating and semi-alternating conditions from the recent paper [6] in order to state our main result.…”

“…The notion of pattern avoidance is exactly as in [1,2]. , (2,4), (3,6), (4,5), (5,2), (6, 1)} of Y = (6 4 , 5, 4) contains M (231) because the restriction of T to the yellow columns and the pink rows rows is a copy of M (231) in T . We require that X ∈ Y .…”

On Wilf Equivalence for Alternating Permutations

“…In order to prove Theorem 1.4, we also need the following Wilf equivalence for alternating permutations, which was proved by Gowravaram and Jagadeesan [7]. Note that Theorem 2.9 can also be proved by similar reasoning as in the proof of Theorem 2.7.…”

“…We obtain the following non-trivial Wilf equivalence for alternating permutations. Note that Theorems 1.3 through 1.6 were proved by Gowravaram and Jagadeesan [7] for k = 2, 3.…”

“…In this section, we follow the approach given in [1], [2], [3] and [7]. We study pattern avoidance for slightly more general objects than alternating permutation, namely, transversals of alternating Young diagrams.…”