Alternating Permutations with Restrictions and Standard Young Tableaux

Abstract

In this paper, we establish 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 show that 4123-avoiding up-down alternating permutations of length $2n+1$  are in one-to-one correspondence with  standard Young tableaux of shape $(n+1,n,n-1)$, and 4123-avoiding up-down alternating permutations of length $2n$ are in bijection with shifted standard Young tableaux of shape $(n+2,n,n-2)$.