Catalan numbers and power laws in cellular automaton rule 14

TL;DR

This paper analyzes a specific elementary cellular automaton, rule 14, revealing how its density of ones decays as a power law and deriving formulas involving Catalan numbers for preimage counts and probabilities.

Contribution

It introduces explicit formulas for preimages and densities in rule 14 using Catalan numbers, connecting cellular automaton patterns to combinatorial mathematics.

Findings

Density of ones decays as a power law with iterations

Derived explicit formulas for preimages of blocks of length 3

Connected cellular automaton behavior to Catalan numbers

Abstract

We discuss example of an elementary cellular automaton for which the density of ones decays toward its limiting value as a power of the number of iterations $n$. Using the fact that this rule conserves the number of blocks 10 and that preimages of some other blocks exhibit patterns closely related to patterns observed in rule 184, we derive expressions for the number of $n$-step preimages of all blocks of length 3. These expressions involve Catalan numbers, and together with basic properties of iterated probability measures they allow us to to compute the density of ones after $n$ iterations, as well as probabilities of occurrence of arbitrary block of length smaller or equal to 3.