Growth diagram description for the DI_n^k bijection

Develop a growth diagram description of the Halverson–Lewandowski deletion–insertion bijection DI_n^k that maps an integer sequence in [n]^k to a pair consisting of a standard Young tableau P_λ(i) and an n-vacillating tableau Γ_λ(i) of the same shape, thereby providing a growth-diagram-based formulation equivalent to the alternating jeu de taquin and RSK insertion process defining DI_n^k.

Background

The bijection DI_n^k of Halverson and Lewandowski iteratively alternates jeu de taquin deletions and RSK row insertions to map an integer sequence in [n]^k to a pair of tableaux of the same shape: a standard Young tableau and an n-vacillating tableau. While growth diagram methods provide powerful local-rule descriptions for many combinatorial insertion algorithms and have been used extensively for related objects (e.g., simplified vacillating tableaux, arc diagrams, and fillings of Ferrers shapes), an analogous formulation for DI_n^k has not yet been established.

A growth diagram description would likely yield structural insight and facilitate further combinatorial analysis of n-vacillating tableaux beyond the parameter regime n≥2k, paralleling known growth-diagram frameworks for RSK-type correspondences.