Sequences with long range exclusions

TL;DR

This paper investigates the structure and size of sequence sets with long-range exclusion constraints, analyzing their properties through graph theory and focusing on polynomial functions for sequence generation.

Contribution

It reviews existing results on sequence restrictions with long-range exclusions and explores the detailed structure and maximal length of sequences when the exclusion function is polynomial.

Findings

Sequence sets are characterized by graph-theoretic formulations.

The maximal length of sequences with polynomial exclusion functions is highly random asymptotically.

The paper provides a comprehensive overview of known results for various functions and alphabet sizes.

Abstract

Given an alphabet $S$, we consider the size of the subsets of the full sequence space $S^{\rm {\bf Z}}$ determined by the additional restriction that $x_i\not=x_{i+f(n)},\ i\in {\rm {\bf Z}},\ n\in {\rm {\bf N}}.$ Here $f$ is a positive, strictly increasing function. We review an other, graph theoretic, formulation and then the known results covering various combinations of $f$ and the alphabet size. In the second part of the paper we turn to the fine structure of the allowed sequences in the particular case where $f$ is a suitable polynomial. The generation of sequences leads naturally to consider the problem of their maximal length, which turns out highly random asymptotically in the alphabet size.