Onion De Bruijn Sequences: Fixed-Window Counting by Growing the Alphabet

Abstract

We study a fixed-window counting system in which integers are represented by words of constant length while the alphabet grows as needed. This viewpoint arises from De Bruijn sequences: for fixed order $n$, the reverse prefer-max sequence is compatible with alphabet growth, since for each $k$ its restriction to $[k]^n$ is a De Bruijn sequence, yielding an infinite sequence over $\mathbb{N}$. We formalize this through the notion of an onion De Bruijn sequence, prove the resulting structural properties, and count compatible finite onion prefixes by an explicit product formula. For orders $n=2,3$, we give explicit rank and unrank formulas and describe addition and multiplication via finite normalization, with exact carry counts and linear carry complexity in the input layers.