On the Number of Rich Words

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) <= c1.605(n) for some constant c. In addition, Guo, Shallit and Shur conjectured that the number of rich words grows slightly slower than n(root n). We prove that lim (n -> infinity) (n)root R-q(n) = 1 for any q, i.e. R-q(n) has a subexponential growth on any alphabet.