Phase Transitions in Weighted Growth Measures on the Young Graph

We introduce a path-dependent weighted growth measure on the Young graph, obtained byassigning an exponential penalty to row-increasing steps along growth histories of partitions.For fixed size n, this induces a one-parameter family of weights on partitions whose maximizerchanges discontinuously as the penalty parameter varies. We formalize the model, provethat the identity of the maximally weighted partition is piecewise constant in the penaltyparameter with finitely many transition points, and report explicit finite-size transitions forn = 8 computed by dynamic programming. The observed dominance chain (4, 2, 1, 1) →(5, 2, 1) → (6, 1, 1) → (7, 1) → (8) illustrates a sharp interpolation between entropy-drivenbalanced shapes and geometrically simple single-row shapes.