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 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 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.

Pattern avoidance of generalized permutations

“…Indeed, from the proof of Theorem 4.3, we will see that the map θ can be realized by the following algorithm which extends those constructions by Lewis [9], Mei and Wang [12]. and Q = 1 1 1 5 2 2 4 3 5 .…”

“…, j k satisfy τ i 1 j 1 ≤ τ i 2 j 2 ≤ · · · ≤ τ i k j k . When β = (1 α i ), each member of S k αβ is a block-ascending permutation which was first studied in [9].…”

“…Theorem 5.1. [9] If N = kn, there are two bijections between • S k+1 (k n )(1 N ) and the set of SYTs of shape (k n ),…”

“…In 2011, Lewis [9] established a bijection between block-ascending permutations and rectangular SYTs to enumerate two classes of block-ascending permutations via the celebrated hook length formula. In Section 4, Lewis's construction will be extended to any pair of SSYTs.…”

“…In Section 5, we present some applications of above results which contain the main results of Lewis [9], Mei and Wang [12], and Chen [7].…”