On the minimum density of monotone subwords

TL;DR

This paper investigates the minimal density of monotone subwords in words over finite alphabets, providing exact values for small cases, conjectures for general cases, and asymptotic behavior as alphabet size grows.

Contribution

It determines exact minimal densities for small alphabet sizes and subword lengths, introduces a construction conjectured to be optimal, and analyzes the asymptotic gap between finite and infinite alphabet cases.

Findings

Exact values of f(2,k) for all k.

Determination of f(3,3), which is irrational.

Asymptotic gap between finite and infinite alphabet densities is Θ(1/s).

Abstract

We consider the asymptotic minimum density $f(s,k)$ of monotone $k$-subwords of words over a totally ordered alphabet of size $s$. The unrestricted alphabet case, $f(\infty,k)$, is well-studied, known for $f(\infty,3)$ and $f(\infty,4)$, and, in particular, conjectured to be rational for all $k$. Here we determine $f(2,k)$ for all $k$ and determine $f(3,3)$, which is already irrational. We describe an explicit construction for all $s$ which is conjectured to yield $f(s,3)$. Using our construction and flag algebra, we determine $f(4,3),f(5,3),f(6,3)$ up to $10^{-3}$ yet argue that flag algebra, regardless of computational power, cannot determine $f(5,3)$ precisely. Finally, we prove that for every fixed $k \ge 3$, the gap between $f(s,k)$ and $f(\infty,k)$ is $\Theta(\frac{1}{s})$.