The Number of Distinct Subpalindromes in Random Words

We prove that a random word of length n over a k-ary fixed alphabet contains, on expectation,Θ(√n) distinct palindromic factors. We study this number of factors, E(n, k), in detail, showing that the limit limn→∞ E(n, k)=√n does not exist for any κ ≥ 2, lim infn→∞ E(n; k)=√n =Θ(1), and lim supn→∞ E(n; k)=√n = Θ(√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.

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Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).