The Number of Distinct Subpalindromes in Random Words

TL;DR

This paper analyzes the expected number of distinct palindromic factors in random words, revealing complex asymptotic behavior and differences between even and odd length palindromes, supported by theoretical proofs and experimental data.

Contribution

It provides a detailed asymptotic analysis of the expected number of palindromic factors in random words, highlighting non-convergent behavior and the impact of palindrome length parity.

Findings

Expected number of palindromic factors is Θ(√n)

Limit behavior of E(n,k)/√n is non-convergent for k≥2

Asymmetry between even and odd length palindromes affects counts

Abstract

We prove that a random word of length $n$ over a $k$-ary fixed alphabet contains, on expectation, $\Theta(\sqrt{n})$ distinct palindromic factors. We study this number of factors, $E(n,k)$, in detail, showing that the limit $\lim_{n\to\infty}E(n,k)/\sqrt{n}$ does not exist for any $k\ge2$, $\liminf_{n\to\infty}E(n,k)/\sqrt{n}=\Theta(1)$, and $\limsup_{n\to\infty}E(n,k)/\sqrt{n}=\Theta(\sqrt{k})$. Such a complicated behaviour stems from the asymmetry between the palindromes of even and odd length. We show that a similar, but much simpler, result on the expected number of squares in random words holds. We also provide some experimental data on the number of palindromic factors in random words.