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

TL;DR

This paper introduces a new class of permutations generalizing alternating permutations, establishes bijections with Young tableaux avoiding certain patterns, and extends pattern avoidance results to skew shapes, providing new enumeration formulas.

Contribution

It defines the class L_{n,k} of permutations, proves pattern-avoidance bijections with Young tableaux, and extends pattern avoidance analysis to skew shapes and reading words.

Findings

Bijections between pattern-avoiding alternating permutations and Young tableaux.

First enumeration of alternating permutations avoiding length four patterns.

Pattern avoidance in reading words of skew shape tableaux relates to Catalan numbers.

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 two bijections between the set A_{2n}(1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape (3^n), and between the set A_{2n + 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 lambda/mu whose reading words avoid 213 is a natural mu-analogue of the Catalan numbers (and in particular does not depend on lambda, up to a simple technical condition), and that there are similar results for the patterns 132, 231 and 312.