Alternating permutations with restrictions and standard Young tableaux

TL;DR

This paper establishes bijections between certain pattern-avoiding alternating permutations and standard Young tableaux, providing new enumeration methods for these combinatorial objects.

Contribution

It introduces novel bijections linking 4123-avoiding alternating permutations with standard Young tableaux and their shifted variants, enabling enumeration of these permutations.

Findings

Bijections between 4123-avoiding down-up alternating permutations and standard Young tableaux.

Enumeration formulas for 4123-avoiding up-down alternating permutations.

Connections established via Yamanouchi words and shifted tableaux.

Abstract

In this paper, we give bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating permutations of length $2n-1$ and the set of shifted standard Young tableaux of shape $(n+1, n, n-1)$ via an intermediate structure of Yamanouchi words. Moreover, we get the enumeration of 4123-avoiding up-down alternating permutations of even and odd length by presenting bijections between 4123-avoiding up-down alternating permutations and standard Young tableaux.