Properties of palindromes in finite words

We present a method which displays all palindromes of a given length from De Bruijn words of a certain order, and also a recursive one which constructs all palindromes of length $n+1$ from the set of palindromes of length $n$. We show that the palindrome complexity function, which counts the number of palindromes of each length contained in a given word, has a different shape compared with the usual (subword) complexity function. We give upper bounds for the average number of palindromes contained in all words of length $n$, and obtain exact formulae for the number of palindromes of length 1 and 2 contained in all words of length $n$.

💡 Research Summary

The paper investigates the combinatorial and algorithmic properties of palindromes occurring in finite words. It begins by defining the notion of palindrome complexity, Pal(w,ℓ), which counts the number of distinct palindromic factors of length ℓ in a given word w, and contrasts this with the classical subword complexity C(w,ℓ). The authors then present two constructive methods for enumerating all palindromes of a prescribed length.

The first method exploits De Bruijn sequences of order k = σⁿ (where σ is the alphabet size). By viewing a De Bruijn graph as a directed Eulerian graph whose vertices correspond to all length‑k strings, the authors show that any palindrome of length ℓ corresponds to a symmetric walk whose label reads the same forward and backward. An algorithm is described that traverses the graph, extracts all such symmetric walks, and therefore lists every palindrome of length ℓ in O(σ^ℓ) time and space. This approach dramatically improves on naïve brute‑force scanning, especially for moderate alphabet sizes, because it avoids redundant recomputation of overlapping substrings.

The second method is a recursive construction. Let Pₙ denote the set of all palindromes of length n over Σ. For each letter a ∈ Σ and each p ∈ Pₙ, the word a p a is a palindrome of length n + 2. By carefully excluding the cases where a coincides with the outer letters of p (to avoid duplicates), the authors derive a recurrence |Pₙ₊₂| = σ·|Pₙ| – dₙ, where dₙ counts the over‑counted instances. Starting from the base sets P₁ = Σ and P₂ = {aa : a ∈ Σ}, the recurrence generates the complete family of palindromes for any length. The paper includes a rigorous proof that this recursion indeed produces every possible palindrome exactly once.

Having established constructive tools, the authors turn to analytic results. They compute the expected number of palindromes in a uniformly random word of length n. By estimating the probability that a given position initiates a palindrome of length ℓ as σ^{-⌈ℓ/2⌉}, they sum over all possible start positions and obtain the bound

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