Pattern avoidance and RSK-like algorithms for alternating permutations and Young tableaux

We define a class Ln,k of permutations that generalizes alternating (up-down) permutations. We then give bijective proofs of certain pattern-avoidance results for this class. The bijections employed include are a modified form of the RSK insertion algorithm and a different bijection with RSK-like properties. As a special case of our results, we give two bijections between the set A2n(1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape 〈3〉, and between the set A2n+1(1234) and the standard Young tableaux of shape 〈3n−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 Ln,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 μanalogue 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. For the patterns 123 and 321, we show that counting the associated tableaux reduces to the problem of counting 123-avoiding permutations with a given descent set, and this computation is easy to carry out recursively.