On the Correspondence Between Integer Sequences and Vacillating Tableaux

A fundamental identity in the representation theory of the partition algebra is nᵏ = ∑λ f^λ mₖ^λ for n ≥ 2k, where λ ranges over integer partitions of n, f^λ is the number of standard Young tableaux of shape λ, and mₖ^λ is the number of vacillating tableaux of shape λ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection DIₙᵏ that maps each integer sequence in [n]ᵏ to a pair of tableaux of the same shape, where one is a standard Young tableau and the other is a vacillating tableau. In this paper, we study the fine properties of Halverson and Lewandowski's bijection and explore the correspondence between integer sequences and the vacillating tableaux via the map DIₙᵏ for general integers n and k. In particular, we characterize the integer sequences i whose corresponding shape, λ, in the image DIₙᵏ(i), satisfies λ₁ = n or λ₁ = n-k.