Pattern avoidance for alternating permutations and Young tableaux

Abstract

We define a class L n , k of permutations that generalizes alternating (up–down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give bijections between the set A 2 n ( 1234 ) of alternating permutations of length 2 n with no four-term increasing subsequence and standard Young tableaux of shape 〈 3 n 〉 , and between the set A 2 n + 1 ( 1234 ) and standard Young tableaux of shape 〈 3 n − 1 , 2 , 1 〉 . This represents the first enumeration of alternating permutations avoiding a pattern of length four. We also extend previous work on doubly-alternating permutations (alternating permutations whose inverses are alternating) to our more general context. The set L n , k may be viewed as the set of reading words of the standard Young tableaux of a certain skew shape. In the last section of the paper, we expand our study to consider pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape λ / μ whose reading words avoid 213 is a natural μ-analogue of the Catalan numbers (and in particular does not depend on λ, up to a simple technical condition), and that there are similar results for the patterns 132, 231 and 312.