Weighted Event-Growth Sequences: Enumeration and Bijections

We introduce the weighted Event-Growth model, a three-parameter family of restricted-growth words based on three intrinsic relations between a new entry and its history: it may repeat the current value, return to another previously used value, or appear for the first time. These events are called Level, Return, and New, and assigning nonnegative rewards to them determines the future admissible range. Although admissibility depends on the entire prefix, we prove that all continuation multiplicities and state updates are determined by only two integer coordinates. This yields a universal succession rule, a reversible Event-Slot encoding, a common differential system, and an integral formula for the shifted exponential generating functions. Four natural one-parameter families have exponential, Bell, Euler, and rational kernels. We construct direct bijections with colored increasing forests of height at most two, cycle–spider assemblies, partial injections, and Dowling partitions. Return-Growth and Reuse-Growth appear as neighboring points of the Euler family, placing several previously separate restricted-growth rules within a single small-state and bijective framework.