On the Number of Rich Words

  • Josef RukavickaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)


Any finite word w of length n contains at most \(n+1\) distinct palindromic factors. If the bound \(n+1\) is reached, the word w is called rich. The number of rich words of length n over an alphabet of cardinality q is denoted \(R_q(n)\). For binary alphabet, Rubinchik and Shur deduced that \({R_2(n)}\le c 1.605^n \) for some constant c. In addition, Guo, Shallit and Shur conjectured that the number of rich words grows slightly slower than \(n^{\sqrt{n}}\). We prove that \(\lim \limits _{n\rightarrow \infty }\root n \of {R_q(n)}=1\) for any q, i.e. \(R_q(n)\) has a subexponential growth on any alphabet.


Rich words Enumeration Palindromes Palindromic factorization 



The author wishes to thank Edita Pelantová and Štěpán Starosta for their useful comments. The author acknowledges support by the Czech Science Foundation grant GAČR 13-03538S and by the Grant Agency of the Czech Technical University in Prague, grant No. SGS14/205/OHK4/3T/14.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePrague 2Czech Republic

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