Upper Bounds for Sequence Saturation
Shihan Kanungo
TL;DR
This work advances the understanding of sequence saturation by presenting an explicit algorithm to build u-saturated sequences and proving linear-time bounds for several families of patterns, including u=abcabc... and u=aa...bb, as well as for a broad class of 3-letter patterns under structural assumptions. It also establishes the existence of doubly infinite u-saturated sequences for many u and offers a linear programming framework to compute Sat(n,u) exactly, providing both theoretical and computational tools. Collectively, the results push toward resolving the general O(n) conjecture for Sat(n,u) and connect saturation concepts to constructive and optimization methods. The paper thus contributes new techniques, structural classifications, and practical algorithms for sequence avoidance and saturation problems.
Abstract
In this paper, we study the saturation function $\mathrm{Sat}(n,u)$ for sequences. Saturation for sequences was introduced by Anand, Geneson, Kaustav, and Tsai (2021), who proved that $\mathrm{Sat}(n,u)=O(n)$ for two-letter sequences $u$ and conjectured that this bound holds for all sequences. We present an algorithm that constructs a $u$-saturated sequence on $n$ letters and apply it to show $\mathrm{Sat}(n,u)=O(n)$ for several families of sequences $u$, including all repetitions of the form $abcabc\dots$. We further establish $\mathrm{Sat}(n,u)=O(n)$ for a broad class of sequences of the form $aa\dots bb$. In addition, we prove that for most sequences $u$, there exists an infinite $u$-saturated sequence. For three-letter sequences of the form $abc\dots xyz$, where $a,b,c$ are distinct and $xyz$ is a permutation of $abc$, we show -- under certain structural assumptions on $u$ -- that $\mathrm{Sat}(n,u)=O(n)$. Finally, we describe a linear program that computes the exact value of $\mathrm{Sat}(n,u)$ for arbitrary $n$ and $u$.