Abstract
We study possible behaviour of the function of prefix palindromic length \(PPL_u(n)\) of an infinite word u, that is, the minimal number of palindromes to which the prefix of length n of u can be decomposed. In a 2013 paper with Puzynina and Zamboni we stated the conjecture that \(PPL_u(n)\) is unbounded for every infinite word u which is not ultimately periodic. Up to now, the conjecture has been proved only for some particular cases including all fixed points of morphisms and, later, Sturmian words.
To give an upper bound for the palindromic length, it is in general sufficient to point out a decomposition of a given word to a given number of palindromes. Proving that such a decomposition does not exist is a trickier question. In this paper, we summarize the existing techniques which can be used for lower bounds on the palindromic length. In particular, we completely describe the prefix palindromic length of the Thue-Morse word and use appropriate numeration systems to give a lower bound for the palindromic length of some Toeplitz words.
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References
Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)
Ambrož, P., Kadlec, O., Masáková, Z., Pelantová, E.: Palindromic length of words and morphisms in class \(\cal{P}\). Preprint. https://arxiv.org/abs/1812.00711
Bardakov, V., Shpilrain, V., Tolstykh, V.: On the palindromic and primitive widths of a free group. J. Algebra 285, 574–585 (2005)
Fici, G., Gagie, T., Kärkkäinen, J., Kempa, D.: A subquadratic algorithm for minimum palindromic factorization. J. Discr. Alg. 28, 41–48 (2014)
Frid, A.E.: Sturmian numeration systems and decompositions to palindromes. Eur. J. Combin. 71, 202–212 (2018)
Frid, A.: Representations of palindromes in the Fibonacci word. In: Proceedings of Numeration, pp. 9–12 (2018)
Frid, A., Puzynina, S., Zamboni, L.: On palindromic factorization of words. Adv. Appl. Math. 50, 737–748 (2013)
Borozdin, K., Kosolobov, D., Rubinchik, M., Shur, A.M.: Palindromic length in linear time. In: CPM 2017, pp. 23:1–23:12 (2017)
Rubinchik, M., Shur, A.M.: EERTREE: an efficient data structure for processing palindromes in strings. Eur. J. Combin. 68, 249–265 (2018)
Saarela, A.: Palindromic length in free monoids and free groups. In: Brlek, S., Dolce, F., Reutenauer, C., Vandomme, É. (eds.) WORDS 2017. LNCS, vol. 10432, pp. 203–213. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66396-8_19
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Frid, A.E. (2019). First Lower Bounds for Palindromic Length. In: Hofman, P., Skrzypczak, M. (eds) Developments in Language Theory. DLT 2019. Lecture Notes in Computer Science(), vol 11647. Springer, Cham. https://doi.org/10.1007/978-3-030-24886-4_17
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DOI: https://doi.org/10.1007/978-3-030-24886-4_17
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