Palindrome complexity versus factor complexity

2026-06-06

AI Summary

This study investigates the asymptotic relationship between palindromic complexity $ \mathrm{Pal}(n) $ and factor complexity $ \rho(n) $ in aperiodic infinite words. Employing combinatorial methods from formal language theory together with asymptotic analysis techniques, the authors establish that $ \frac{\mathrm{Pal}(n)\log(\mathrm{Pal}(n)+1)}{\rho(n)} \to 0 $ as $ n \to \infty $. This result demonstrates the asymptotic sparsity of palindromic factors in aperiodic sequences and yields a novel upper bound relating the two complexity functions. Moreover, the work shows that this upper bound is essentially optimal with respect to the form of its numerator, thereby providing a sharp characterization of the growth disparity between palindromic and general factor complexities in non-periodic infinite words.

Abstract

Let ${\bf x} = (a_i)_{i \geq 0}$ be an infinite word over a finite alphabet $Σ$. Let $ρ(n)$ be the factor complexity function for $\bf x$ and ${\rm Pal}(n)$ be the palindrome complexity function for $\bf x$. We give a new relationship between these two quantities; namely, if $\bf x$ is not ultimately periodic, then $$ \lim_{n \rightarrow \infty} {{ {\rm Pal} (n) \log ({\rm Pal} (n) + 1)} \over {ρ(n)}} = 0. $$ Furthermore, we prove that the numerator in this result is essentially optimal.

Problem

• palindrome complexity
• factor complexity
• infinite words
• ultimately periodic
• asymptotic relation

Innovation

• palindrome complexity
• factor complexity
• infinite words
• asymptotic relation
• combinatorics on words