On the Number of Rich Words

Abstract

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.

Appendix

For the reader’s convenience, we provide a proof of the well-known inequality we used in the proof of Proposition 9.

Lemma 11

$\sum _{k=0}^L\left( {\begin{array}{c}N\\ k\end{array}}\right) \le \left( \frac{\mathrm {e}N}{L}\right) ^L$, where $L\le N$ and $L,N\in \mathbb {Z}^+$ ($\mathbb {Z}^+$ denotes the set of positive integers).

Proof

Consider $x\in (0,1]$. The binomial theorem states that

$$(1+x)^N=\sum _{k=0}^N\left( {\begin{array}{c}N\\ k\end{array}}\right) x^k\ge \sum _{k=0}^L\left( {\begin{array}{c}N\\ k\end{array}}\right) x^k\text{. }$$

By dividing by the factor $x^L$ we obtain

$$ \sum _{k=0}^L\left( {\begin{array}{c}N\\ k\end{array}}\right) x^{k-L}\le \frac{(1+x)^N}{x^L}\text{. }$$

Since $x\in (0,1]$ and $k-L\le 0$, then $x^{k-L}\ge 1$, it follows that

$$ \sum _{k=0}^L\left( {\begin{array}{c}N\\ k\end{array}}\right) \le \frac{(1+x)^N}{x^L}\text{. }$$

Let us substitute $x=\frac{L}{N}\in (0,1]$ and let us use the inequality $1+x<\mathrm {e}^x$, that holds for all $x>0$:

$$ \frac{(1+x)^N}{x^L}\le \frac{\mathrm {e}^{xN}}{x^L}=\frac{\mathrm {e}^{\frac{L}{N}N}}{(\frac{L}{N})^L}=\left( \frac{\mathrm {e}N}{L}\right) ^L \text{. }$$