Subexponential upper bound on the number of rich words

Let $R(n)$ denote the number of rich words of length $n$ over a given finite alphabet. In 2017 it was proved that ${\lim }_{n\to \mathrm{\infty }}\sqrt[n]{R(n)}=1$; it means the number of rich words has a subexponential growth. However, up to now, no subexponential upper bound on $R(n)$ has been presented. The current paper fills this gap. Let $\frac{1}{2}<\lambda <1$ and $\gamma >1$ be real constants, let $q$ be the size of the alphabet, and let $\phi$ be a positive function with ${\lim }_{n\to \mathrm{\infty }}\phi (n)=\mathrm{\infty }$ and ${\lim }_{n\to \mathrm{\infty }}\frac{n}{\phi (n)}=\mathrm{\infty }$. Let ${\ln }^{\ast }(x)$ denote the iterated logarithm of $x>0$. We prove that there are $n_0$ and $c>0$ such that if $n>n_0$, $$f(n)=\sqrt[\gamma ]{c{\ln }^{\ast }(\frac{n}{\phi (n)}\ln q)}andB(n)=q^{\frac{n}{\phi (n)}+\frac{n}{(2\lambda {)}^{f(n)-1}}}\text{mbox}$$ then ${\lim }_{n\to \mathrm{\infty }}\sqrt[n]{B(n)}=1$ and $R(n)\le B(n)$.