What is the idea?

Every entry after the initial zero does exactly one of three things: it repeats the current value (Level), returns to a different value seen earlier (Return), or introduces a value that has never appeared before (New). The model assigns nonnegative rewards a, b and c to those three event types.

The accumulated reward becomes a growth budget: it enlarges the interval of values allowed at later steps. So the same event history can be legal under one reward triple and illegal under another. This turns many apparently different restricted-growth rules into points of one parameter space.

L

Level

Repeat the current value.

R

Return

Use an old value different from the current one.

N

New

Use a value that has not appeared before.

Why is it interesting?

A three-parameter rule could easily have become a catalogue of unrelated cases. Instead, the paper proves a small-state theorem: all future branching is determined by only two integers — the number of Return choices and the number of New choices currently available.

That collapse is the structural payoff. It produces one universal succession rule, a reversible Event-Slot history that remembers ranked choices, and a common differential system for the whole family. Classical objects then appear naturally in special parameter fibers rather than as isolated coincidences.

Return-Growth and Reuse-Growth sit next to each other in the Euler fiber at (0,1,0) and (1,1,0). The separate Reuse-Growth preprint studies the latter point in depth from the pattern-avoidance side.

Read the Reuse-Growth paper →

Main results

01

Two-coordinate branching

For every nonnegative reward triple (a,b,c), the next legal values split into exactly one Level choice, p Return choices and q New choices. Their state updates depend only on (p,q), not on the full word.

02

A reversible Event-Slot encoding

Each legal extension is matched by rank to an active slot of type L, R or N. The full slot history is bijective with the original word and can be inverted by removing the latest time label.

03

One master flow, four familiar kernels

The slot decomposition yields a common differential system and an integral formula for the shifted exponential generating function. The four zero–one (b,c) fibers have exponential, Bell, Euler and rational kernels.

04

Direct bijections to classical structures

Important fibers are mapped directly to colored increasing forests of height at most two, cycle–spider assemblies, partial injections and Dowling partitions.

05

One parameter space for old and new models

The eight zero–one reward vectors form an ordered Boolean core. Ordinary restricted-growth words, level sequences, change sequences, subexcedant sequences, Return-Growth and Reuse-Growth all appear as explicit points in the same framework.

For specialists

Abstract

Restricted-growth words are built by letting a prefix control which values may appear next. We introduce the weighted Event-Growth model, a three-parameter family 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. Assigning nonnegative rewards to these Level, Return, and New events determines the future admissible range. Although admissibility depends on the entire prefix, we prove that all continuation multiplicities and state updates collapse to 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 one small-state and bijective framework.

Key formulas

Growth budgetgₐ,ᵦ,𝑐(w) = aL(w) + bR(w) + cN(w)
Legal rulew₁ = 0, 0 ≤ wᵢ ≤ 1 + gₐ,ᵦ,𝑐(w₁…wᵢ₋₁)
Two-coordinate statep = N, q = g + 1 − N
Succession ruleL: (p,q)→(p,q+a); R: p×(p,q+b); N: q×(p+1,q+c−1)
Master flowU′ = UVᵇ, V′ = UVᶜ, M′ = MVᵃ, H = MV
Four zero–one kernelsV₀₀=eᵗ; V₀₁=e^(eᵗ−1); V₁₀=sec t+tan t; V₁₁=(1−t)⁻¹

Citation / BibTeX

@misc{kleiner2026weightedeventgrowth,
  author = {Igor Kleiner},
  title = {Weighted Event-Growth Sequences: Enumeration and Bijections},
  year = {2026},
  month = sep,
  note = {Preprint},
  doi = {10.5281/zenodo.22920635},
  url = {https://doi.org/10.5281/zenodo.22920635}
}