Property of upper bounds on the number of rich words

TL;DR

This paper establishes upper bounds on the number of rich words, which are words containing the maximum number of palindromic factors, by analyzing functions related to their length and palindromic structure.

Contribution

It introduces new bounds on the count of rich words using functions that relate to their palindromic length and provides conditions under which these bounds hold.

Findings

Derived upper bounds for the number of rich words based on length and palindromic properties.

Established conditions involving functions nd or bounding rich words.

Provided asymptotic behavior of rich words count as length increases.

Abstract

A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. Let $q\geq 2$ be the size of the alphabet. Let $R(n)$ be the number of rich words of length $n$. Let $d>1$ be a real constant and let $\phi, \psi$ be real functions such that \begin{itemize}\item there is $n_0$ such that $2\psi(2^{-1}\phi(n))\geq d\psi(n)$ for all $n>n_0$, \item $\frac{n}{\phi(n)}$ is an upper bound on the palindromic length of rich words of length $n$, and \item $\frac{x}{\psi(x)}+\frac{x\ln{(\phi(x))}}{\phi(x)}$ is a strictly increasing concave function. \end{itemize} We show that if $c_1,c_2$ are real constants and $R(n)\leq q^{c_1\frac{n}{\psi(n)}+c_2\frac{n\ln(\phi(n))}{\phi(n)}}$ then for every real constant $c_3>0$ there is a positive integer $n_0$ such that for all $n>n_0$ we have that \[R(n)\leq q^{(c_1+c_3)\frac{n}{d\psi(n)}+c_2\frac{n\ln(\phi(n))}{\phi(n)}(1+\frac{1}{c_2\ln{q}}+c_3)}\mbox{.}\]