Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux

TL;DR

This paper explores pattern avoidance in alternating permutations and Young tableaux, extending shape-Wilf equivalence and analyzing permutations with restricted ascent-descent patterns, revealing new enumeration relationships.

Contribution

It generalizes shape-Wilf equivalence to alternating permutations and studies pattern avoidance in permutations with specific ascent-descent structures, providing new enumeration results.

Findings

Extended shape-Wilf equivalence to alternating permutations

Determined sizes of pattern-avoiding permutations with ascent-descent restrictions

Established inequalities for pattern-avoiding permutation sets

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, 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.